integral of (81)/(x^4+27x)

{ "query": { "display": "$$\\int\\:\\frac{81}{x^{4}+27x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{81}{x^{4}+27x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{81}{x^{4}+27x}dx=3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|+C$$", "input": "\\int\\:\\frac{81}{x^{4}+27x}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=81\\cdot\\:\\int\\:\\frac{1}{x^{4}+27x}dx" }, { "type": "interim", "title": "Take the partial fraction of $$\\frac{1}{x^{4}+27x}:{\\quad}\\frac{1}{27x}-\\frac{1}{81\\left(x+3\\right)}+\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}$$", "input": "\\frac{1}{x^{4}+27x}", "steps": [ { "type": "interim", "title": "Factor $$x^{4}+27x:{\\quad}x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)$$", "input": "x^{4}+27x", "result": "=\\frac{1}{x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}", "steps": [ { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x^{3}+27\\right)$$", "input": "x^{4}+27x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{4}=x^{3}x$$" ], "result": "=x^{3}x+27x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x^{3}+27\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=x\\left(x^{3}+27\\right)" }, { "type": "interim", "title": "Factor $$x^{3}+27:{\\quad}\\left(x+3\\right)\\left(x^{2}-3x+3^{2}\\right)$$", "input": "x^{3}+27", "steps": [ { "type": "step", "primary": "Rewrite $$27$$ as $$3^{3}$$", "result": "=x^{3}+3^{3}" }, { "type": "step", "primary": "Apply Sum of Cubes Formula: $$x^{3}+y^{3}=\\left(x+y\\right)\\left(x^{2}-xy+y^{2}\\right)$$", "secondary": [ "$$x^{3}+3^{3}=\\left(x+3\\right)\\left(x^{2}-3x+3^{2}\\right)$$" ], "result": "=\\left(x+3\\right)\\left(x^{2}-3x+3^{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Sum%2FDifference%20of%20Cubes", "practiceTopic": "Factor Sum of Cubes" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=x\\left(x+3\\right)\\left(x^{2}-3x+3^{2}\\right)" }, { "type": "step", "primary": "Refine", "result": "=x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Create the partial fraction template using the denominator $$x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)$$", "result": "\\frac{1}{x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}=\\frac{a}{x}+\\frac{b}{x+3}+\\frac{a_{3}x+a_{2}}{x^{2}-3x+9}", "steps": [ { "type": "step", "primary": "For $$x\\:$$add the partial fraction(s): $$\\frac{a}{x}$$" }, { "type": "step", "primary": "For $$x+3\\:$$add the partial fraction(s): $$\\frac{b}{x+3}$$" }, { "type": "step", "primary": "For $$x^{2}-3x+9\\:$$add the partial fraction(s): $$\\frac{a_{3}x+a_{2}}{x^{2}-3x+9}$$" }, { "type": "step", "result": "\\frac{1}{x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}=\\frac{a}{x}+\\frac{b}{x+3}+\\frac{a_{3}x+a_{2}}{x^{2}-3x+9}" } ], "meta": { "interimType": "Partial Fraction Templates Top 1Eq" } }, { "type": "step", "primary": "Multiply equation by the denominator", "result": "\\frac{1\\cdot\\:x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}{x\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}=\\frac{ax\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}{x}+\\frac{bx\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}{x+3}+\\frac{x\\left(a_{3}x+a_{2}\\right)\\left(x+3\\right)\\left(x^{2}-3x+9\\right)}{x^{2}-3x+9}" }, { "type": "step", "primary": "Simplify", "result": "1=a\\left(x+3\\right)\\left(x^{2}-3x+9\\right)+bx\\left(x^{2}-3x+9\\right)+x\\left(a_{3}x+a_{2}\\right)\\left(x+3\\right)" }, { "type": "step", "primary": "Solve the unknown parameters by plugging the real roots of the denominator: $$0,\\:-3$$" }, { "type": "interim", "title": "For the denominator root $$0:{\\quad}a=\\frac{1}{27}$$", "steps": [ { "type": "step", "primary": "Plug in $$x=0\\:$$into the equation", "result": "1=a\\left(0+3\\right)\\left(0^{2}-3\\cdot\\:0+9\\right)+b\\cdot\\:0\\cdot\\:\\left(0^{2}-3\\cdot\\:0+9\\right)+\\left(a_{3}\\cdot\\:0+a_{2}\\right)\\cdot\\:0\\cdot\\:\\left(0+3\\right)" }, { "type": "step", "primary": "Expand", "result": "1=27a" }, { "type": "interim", "title": "Solve $$1=27a\\:$$for $$a:{\\quad}a=\\frac{1}{27}$$", "input": "1=27a", "result": "a=\\frac{1}{27}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "27a=1" }, { "type": "interim", "title": "Divide both sides by $$27$$", "input": "27a=1", "result": "a=\\frac{1}{27}", "steps": [ { "type": "step", "primary": "Divide both sides by $$27$$", "result": "\\frac{27a}{27}=\\frac{1}{27}" }, { "type": "step", "primary": "Simplify", "result": "a=\\frac{1}{27}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 2Eq" } } ], "meta": { "interimType": "Partial Fraction Single Root 1Eq" } }, { "type": "interim", "title": "For the denominator root $$-3:{\\quad}b=-\\frac{1}{81}$$", "steps": [ { "type": "step", "primary": "Plug in $$x=-3\\:$$into the equation", "result": "1=a\\left(\\left(-3\\right)+3\\right)\\left(\\left(-3\\right)^{2}-3\\left(-3\\right)+9\\right)+b\\left(-3\\right)\\left(\\left(-3\\right)^{2}-3\\left(-3\\right)+9\\right)+\\left(a_{3}\\left(-3\\right)+a_{2}\\right)\\left(-3\\right)\\left(\\left(-3\\right)+3\\right)" }, { "type": "step", "primary": "Expand", "result": "1=-81b" }, { "type": "interim", "title": "Solve $$1=-81b\\:$$for $$b:{\\quad}b=-\\frac{1}{81}$$", "input": "1=-81b", "result": "b=-\\frac{1}{81}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-81b=1" }, { "type": "interim", "title": "Divide both sides by $$-81$$", "input": "-81b=1", "result": "b=-\\frac{1}{81}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-81$$", "result": "\\frac{-81b}{-81}=\\frac{1}{-81}" }, { "type": "step", "primary": "Simplify", "result": "b=-\\frac{1}{81}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 2Eq" } } ], "meta": { "interimType": "Partial Fraction Single Root 1Eq" } }, { "type": "step", "result": "a=\\frac{1}{27},\\:b=-\\frac{1}{81}" }, { "type": "step", "primary": "Plug in the solutions to the known parameters", "result": "1=\\frac{1}{27}\\left(x+3\\right)\\left(x^{2}-3x+9\\right)+\\left(-\\frac{1}{81}\\right)x\\left(x^{2}-3x+9\\right)+x\\left(a_{3}x+a_{2}\\right)\\left(x+3\\right)" }, { "type": "step", "primary": "Expand", "result": "1=a_{3}x^{3}+3a_{3}x^{2}+a_{2}x^{2}+3a_{2}x+\\frac{2x^{3}}{81}+\\frac{x^{2}}{27}-\\frac{x}{9}+1" }, { "type": "step", "primary": "Extract Variables from within fractions", "result": "1=a_{3}x^{3}+3a_{3}x^{2}+a_{2}x^{2}+3a_{2}x+\\frac{2}{81}x^{3}+\\frac{1}{27}x^{2}-\\frac{1}{9}x+1" }, { "type": "step", "primary": "Group elements according to powers of $$x$$", "result": "1=x^{3}\\left(a_{3}+\\frac{2}{81}\\right)+x^{2}\\left(a_{2}+3a_{3}+\\frac{1}{27}\\right)+x\\left(3a_{2}-\\frac{1}{9}\\right)+1" }, { "type": "step", "primary": "Equate the coefficients of similar terms on both sides to create a list of equations", "result": "\\begin{bmatrix}3a_{2}-\\frac{1}{9}=0\\\\3a_{3}+a_{2}+\\frac{1}{27}=0\\\\a_{3}+\\frac{2}{81}=0\\end{bmatrix}" }, { "type": "interim", "title": "Solve system of equations:$${\\quad}a_{2}=\\frac{1}{27},\\:a_{3}=-\\frac{2}{81}$$", "result": "a_{2}=\\frac{1}{27},\\:a_{3}=-\\frac{2}{81}", "steps": [ { "type": "step", "result": "\\begin{bmatrix}3a_{2}-\\frac{1}{9}=0\\\\3a_{3}+a_{2}+\\frac{1}{27}=0\\\\a_{3}+\\frac{2}{81}=0\\end{bmatrix}" }, { "type": "interim", "title": "Isolate $$a_{2}\\:$$for $$3a_{2}-\\frac{1}{9}=0:{\\quad}a_{2}=\\frac{1}{27}$$", "input": "3a_{2}-\\frac{1}{9}=0", "steps": [ { "type": "interim", "title": "Move $$\\frac{1}{9}\\:$$to the right side", "input": "3a_{2}-\\frac{1}{9}=0", "result": "3a_{2}=\\frac{1}{9}", "steps": [ { "type": "step", "primary": "Add $$\\frac{1}{9}$$ to both sides", "result": "3a_{2}-\\frac{1}{9}+\\frac{1}{9}=0+\\frac{1}{9}" }, { "type": "step", "primary": "Simplify", "result": "3a_{2}=\\frac{1}{9}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3a_{2}=\\frac{1}{9}", "result": "a_{2}=\\frac{1}{27}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3a_{2}}{3}=\\frac{\\frac{1}{9}}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3a_{2}}{3}=\\frac{\\frac{1}{9}}{3}", "result": "a_{2}=\\frac{1}{27}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3a_{2}}{3}:{\\quad}a_{2}$$", "input": "\\frac{3a_{2}}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=a_{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7S9gsdReSu15zhFg0HWxhcxZsnww+P7XjdokU4IcmGg+jkVi15I8rBefLi4Iyt2wr56yZiBxGOlKA4+q9WkPEQYsaY2cs9YZ/08feWmaOgkgpmw6HumtOtBdZArFlLt0Qsu5FY87vo829//0lhqDo1w==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{9}}{3}:{\\quad}\\frac{1}{27}$$", "input": "\\frac{\\frac{1}{9}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{9\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$9\\cdot\\:3=27$$", "result": "=\\frac{1}{27}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajekwdeCO8zB4pMs9Z2t2R1sDnzlbPZjyKgy1eUCFsLd5xXDODtFBCC8Uf836IcE9x3eB/QjQos5wNUsD/0kW3ZnuQCM/vqpbrqU5SxRHBPSd6Tt2GVo3d2lEw7lIGwr7lDmJtMQbJQDX3hrn3+U7n8g=" } }, { "type": "step", "result": "a_{2}=\\frac{1}{27}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 2Eq" } }, { "type": "step", "primary": "Substitute $$a_{2}=\\frac{1}{27}$$", "result": "\\begin{bmatrix}a_{3}+\\frac{2}{81}=0\\\\3a_{3}+\\frac{1}{27}+\\frac{1}{27}=0\\end{bmatrix}" }, { "type": "interim", "title": "Simplify", "input": "3a_{3}+\\frac{1}{27}+\\frac{1}{27}=0", "steps": [ { "type": "interim", "title": "Simplify $$3a_{3}+\\frac{1}{27}+\\frac{1}{27}:{\\quad}3a_{3}+\\frac{2}{27}$$", "input": "3a_{3}+\\frac{1}{27}+\\frac{1}{27}", "steps": [ { "type": "interim", "title": "Combine the fractions $$\\frac{1}{27}+\\frac{1}{27}:{\\quad}\\frac{2}{27}$$", "result": "=3a_{3}+\\frac{2}{27}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1+1}{27}" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{2}{27}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "3a_{3}+\\frac{2}{27}=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\begin{bmatrix}a_{3}+\\frac{2}{81}=0\\\\3a_{3}+\\frac{2}{27}=0\\end{bmatrix}" }, { "type": "interim", "title": "Isolate $$a_{3}\\:$$for $$a_{3}+\\frac{2}{81}=0:{\\quad}a_{3}=-\\frac{2}{81}$$", "input": "a_{3}+\\frac{2}{81}=0", "steps": [ { "type": "interim", "title": "Move $$\\frac{2}{81}\\:$$to the right side", "input": "a_{3}+\\frac{2}{81}=0", "result": "a_{3}=-\\frac{2}{81}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{2}{81}$$ from both sides", "result": "a_{3}+\\frac{2}{81}-\\frac{2}{81}=0-\\frac{2}{81}" }, { "type": "step", "primary": "Simplify", "result": "a_{3}=-\\frac{2}{81}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 2Eq" } }, { "type": "step", "primary": "Substitute $$a_{3}=-\\frac{2}{81}$$", "result": "\\begin{bmatrix}3\\left(-\\frac{2}{81}\\right)+\\frac{2}{27}=0\\end{bmatrix}" }, { "type": "interim", "title": "Simplify", "input": "3\\left(-\\frac{2}{81}\\right)+\\frac{2}{27}=0", "steps": [ { "type": "interim", "title": "$$3\\left(-\\frac{2}{81}\\right)+\\frac{2}{27}=0$$", "input": "3\\left(-\\frac{2}{81}\\right)+\\frac{2}{27}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-3\\cdot\\:\\frac{2}{81}+\\frac{2}{27}" }, { "type": "interim", "title": "$$3\\cdot\\:\\frac{2}{81}=\\frac{2}{27}$$", "input": "3\\cdot\\:\\frac{2}{81}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2\\cdot\\:3}{81}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{6}{81}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{2}{27}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA4I9FAqaIg8xcgeZkzKJvScJQJZuTAY5js+oqjdT8kslwKVVRM+Tw88po6Yv9keG7f8//6/nV5O4fb8Xgwi7mapx4nUY00HlVZKlUcHy0lFDBFqVxcjVybIHjacpoKRS6h7IM83uDaexJaTO81cKvfw=" } }, { "type": "step", "result": "=-\\frac{2}{27}+\\frac{2}{27}" }, { "type": "step", "primary": "Add similar elements: $$-\\frac{2}{27}+\\frac{2}{27}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7T+3vtL6QAqmmpAu+uMu+uUQv4RXh4QM5BJ1nkRUm9Z58kR7hsO/rTOTBE0w4+r1RJAg2NCSHXL2Ksh60M5Oy9WYGynyltByXt5a4b+QbEK1NcxzCHghOro6Ui1Lke8sw8uc9JslMP7NBseJZYXQKnrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "0=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\begin{bmatrix}0=0\\end{bmatrix}" }, { "type": "step", "primary": "The solutions to the system of equations are:", "result": "a_{2}=\\frac{1}{27},\\:a_{3}=-\\frac{2}{81}" } ], "meta": { "solvingClass": "System of Equations", "interimType": "Partial Fraction Solve System Equation 0Eq" } }, { "type": "step", "primary": "Plug the solutions to the partial fraction parameters to obtain the final result", "result": "\\frac{\\frac{1}{27}}{x}+\\frac{\\left(-\\frac{1}{81}\\right)}{x+3}+\\frac{\\left(-\\frac{2}{81}\\right)x+\\frac{1}{27}}{x^{2}-3x+9}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\frac{1}{27}}{x}+\\frac{\\left(-\\frac{1}{81}\\right)}{x+3}+\\frac{\\left(-\\frac{2}{81}\\right)x+\\frac{1}{27}}{x^{2}-3x+9}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{27}}{x}:{\\quad}\\frac{1}{27x}$$", "input": "\\frac{\\frac{1}{27}}{x}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{27x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajcXCoNCp0uSfcEbmOi30OvPNGoPE9TME3q+OPmgkv2RQiEw6G4T+RFI2ZfZDoB3kMrtfSRp9NevSEnhJZDepRF2LGmNnLPWGf9PH3lpmjoJIT6Wk1j0nJ3Nx9hn2Bb6EBhgosg8qJIIkoMQ9PmESTkqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Simplify $$\\frac{\\left(-\\frac{1}{81}\\right)}{x+3}:{\\quad}-\\frac{1}{81\\left(x+3\\right)}$$", "input": "\\frac{-\\frac{1}{81}}{x+3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{1}{81}}{x+3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{1}{81}}{x+3}=\\frac{1}{81\\left(x+3\\right)}$$" ], "result": "=-\\frac{1}{81\\left(x+3\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVYP38YWTzT5AqSz7HtoSS4TehkKrn0era9rz8TlL+x/vttdvQxZI3PlVepHWO3+Ugr5UdpamY5roW9wZ9ol2jaxFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qkZnOV0a0BOvVUL49orBh2J54tRyEQuPWuEzWhoEaYV8" } }, { "type": "interim", "title": "Simplify $$\\frac{\\left(-\\frac{2}{81}\\right)x+\\frac{1}{27}}{x^{2}-3x+9}:{\\quad}\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}$$", "input": "\\frac{\\left(-\\frac{2}{81}\\right)x+\\frac{1}{27}}{x^{2}-3x+9}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-\\frac{2}{81}x+\\frac{1}{27}}{x^{2}-3x+9}" }, { "type": "interim", "title": "Multiply $$\\frac{2}{81}x\\::{\\quad}\\frac{2x}{81}$$", "input": "\\frac{2}{81}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2x}{81}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{-\\frac{2x}{81}+\\frac{1}{27}}{x^{2}-3x+9}" }, { "type": "interim", "title": "Join $$-\\frac{2x}{81}+\\frac{1}{27}:{\\quad}\\frac{-2x+3}{81}$$", "input": "-\\frac{2x}{81}+\\frac{1}{27}", "result": "=\\frac{\\frac{-2x+3}{81}}{x^{2}-3x+9}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$81,\\:27:{\\quad}81$$", "input": "81,\\:27", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$81:{\\quad}3\\cdot\\:3\\cdot\\:3\\cdot\\:3$$", "input": "81", "steps": [ { "type": "step", "primary": "$$81\\:$$divides by $$3\\quad\\:81=27\\cdot\\:3$$", "result": "=3\\cdot\\:27" }, { "type": "step", "primary": "$$27\\:$$divides by $$3\\quad\\:27=9\\cdot\\:3$$", "result": "=3\\cdot\\:3\\cdot\\:9" }, { "type": "step", "primary": "$$9\\:$$divides by $$3\\quad\\:9=3\\cdot\\:3$$", "result": "=3\\cdot\\:3\\cdot\\:3\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRtXnBik8vDPXVw+nKWp28DI6vjYOhFriWa+1b5dMGPIpB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEAaxYN461mup4qR4y5ay+HK" } }, { "type": "interim", "title": "Prime factorization of $$27:{\\quad}3\\cdot\\:3\\cdot\\:3$$", "input": "27", "steps": [ { "type": "step", "primary": "$$27\\:$$divides by $$3\\quad\\:27=9\\cdot\\:3$$", "result": "=3\\cdot\\:9" }, { "type": "step", "primary": "$$9\\:$$divides by $$3\\quad\\:9=3\\cdot\\:3$$", "result": "=3\\cdot\\:3\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFwMjDtSA3s/rRhs6Fmef9mzB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIED0W1khYkn4OtT1XV180RCy" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$81$$ or $$27$$", "result": "=3\\cdot\\:3\\cdot\\:3\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3\\cdot\\:3\\cdot\\:3=81$$", "result": "=81" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$81$$" }, { "type": "step", "primary": "For $$\\frac{1}{27}:\\:$$multiply the denominator and numerator by $$3$$", "result": "\\frac{1}{27}=\\frac{1\\cdot\\:3}{27\\cdot\\:3}=\\frac{3}{81}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=-\\frac{2x}{81}+\\frac{3}{81}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-2x+3}{81}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fKlREPZ8+xcP7ahexfiv7MjsD+xwFGiWLwPn2p718L9hXJgw+bquqWJyTyU9PkDxTpIm2o4PGIuy/WI6oYthZ3CQoYlYQ8U+Tfyx0kyzI8iAvgf+5ZTHHCOprjIXUlhjWxgBHUboaucACrycigui5GRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM96VkRmTgkO+T5G3tM7EYsocjsD+xwFGiWLwPn2p718L9hXJgw+bquqWJyTyU9PkDxzAZ7714EWi6Vw8B+Qm6baQ==" } }, { "type": "step", "result": "=\\frac{1}{27x}-\\frac{1}{81\\left(x+3\\right)}+\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}" } ], "meta": { "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\frac{1}{27x}-\\frac{1}{81\\left(x+3\\right)}+\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}" } ], "meta": { "solvingClass": "Partial Fractions", "interimType": "Algebraic Manipulation Partial Fraction Top Title 1Eq" } }, { "type": "step", "result": "=81\\cdot\\:\\int\\:\\frac{1}{27x}-\\frac{1}{81\\left(x+3\\right)}+\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}dx" }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=81\\left(\\int\\:\\frac{1}{27x}dx-\\int\\:\\frac{1}{81\\left(x+3\\right)}dx+\\int\\:\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}dx\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{27x}dx=\\frac{1}{27}\\ln\\left|x\\right|$$", "input": "\\int\\:\\frac{1}{27x}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{27}\\cdot\\:\\int\\:\\frac{1}{x}dx" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{x}dx=\\ln\\left(\\left|x\\right|\\right)$$", "result": "=\\frac{1}{27}\\ln\\left|x\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{81\\left(x+3\\right)}dx=\\frac{1}{81}\\ln\\left|x+3\\right|$$", "input": "\\int\\:\\frac{1}{81\\left(x+3\\right)}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{81}\\cdot\\:\\int\\:\\frac{1}{x+3}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{x+3}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=x+3$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x+3\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}+\\frac{d}{dx}\\left(3\\right)" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3\\right)=0$$", "input": "\\frac{d}{dx}\\left(3\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYu6nPER/cBcxgb/Kz63vQV1J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTuwXg0Wd+I5tymlezl5JoPF" } }, { "type": "step", "result": "=1+0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=1du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77yO3EBEmJVox+iHngWIUEwcjlLRK1jUV206qo4+vRN7yRnKSRJBHsnEXq1wS/zww7h8KUX9hN21wJE16H/NfZx2cv65xlj4FWO/jAv7Am1CptFvUOUfgDrM9m4ow9eu1Xql8XXPq6bNQlMm+36iNhkkjuzIgeJUg10ybKgq0r22txEId7lZcSHdTAsAvmTZFg==" } }, { "type": "step", "result": "=\\frac{1}{81}\\cdot\\:\\int\\:\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=\\frac{1}{81}\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=x+3$$", "result": "=\\frac{1}{81}\\ln\\left|x+3\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}dx=-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "\\int\\:\\frac{-2x+3}{81\\left(x^{2}-3x+9\\right)}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{81}\\cdot\\:\\int\\:\\frac{-2x+3}{x^{2}-3x+9}dx" }, { "type": "interim", "title": "Expand $$\\frac{-2x+3}{x^{2}-3x+9}:{\\quad}-\\frac{2x}{x^{2}-3x+9}+\\frac{3}{x^{2}-3x+9}$$", "input": "\\frac{-2x+3}{x^{2}-3x+9}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "result": "=-\\frac{2x}{x^{2}-3x+9}+\\frac{3}{x^{2}-3x+9}" } ], "meta": { "interimType": "Algebraic Manipulation Expand 1Eq" } }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\frac{1}{81}\\left(-\\int\\:\\frac{2x}{x^{2}-3x+9}dx+\\int\\:\\frac{3}{x^{2}-3x+9}dx\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{2x}{x^{2}-3x+9}dx=\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$", "input": "\\int\\:\\frac{2x}{x^{2}-3x+9}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=2\\cdot\\:\\int\\:\\frac{x}{x^{2}-3x+9}dx" }, { "type": "interim", "title": "Complete the square $$x^{2}-3x+9:{\\quad}\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}$$", "input": "x^{2}-3x+9", "steps": [ { "type": "step", "primary": "Write $$x^{2}-3x+9\\:$$in the form: $$x^2+2ax+a^2$$" }, { "type": "interim", "title": "$$2a=-3{\\quad:\\quad}a=-\\frac{3}{2}$$", "input": "2a=-3", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-3", "result": "a=-\\frac{3}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-3}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=-\\frac{3}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-\\frac{3}{2}\\right)^{2}\\:$$", "result": "=x^{2}-3x+9+\\left(-\\frac{3}{2}\\right)^{2}-\\left(-\\frac{3}{2}\\right)^{2}" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-3x+\\left(-\\frac{3}{2}\\right)^{2}=\\left(x-\\frac{3}{2}\\right)^{2}$$", "Complete the square" ], "result": "=\\left(x-\\frac{3}{2}\\right)^{2}+9-\\left(-\\frac{3}{2}\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "=\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "=2\\cdot\\:\\int\\:\\frac{x}{\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{x}{\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=x-\\frac{3}{2}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x-\\frac{3}{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(\\frac{3}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{3}{2}\\right)=0$$", "input": "\\frac{d}{dx}\\left(\\frac{3}{2}\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnHXdJ1FY75umoXux1bL9K944OmnsvrgbNkOIEFbFW8WYAIXwDG7/aP+CNC9gUVzaHNS9SX5M3gDB/Er/MAH1V+5QV7agSZLIzF7D9vX0CHvUmusDuyFta1vRswttwqLuLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=1du$$" }, { "type": "step", "result": "=\\int\\:\\frac{x}{u^{2}+\\frac{27}{4}}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{4x}{4u^{2}+27}du" }, { "type": "interim", "title": "$$u=x-\\frac{3}{2}\\quad\\Rightarrow\\quad\\:x=u+\\frac{3}{2}$$", "input": "x-\\frac{3}{2}=u", "steps": [ { "type": "interim", "title": "Move $$\\frac{3}{2}\\:$$to the right side", "input": "x-\\frac{3}{2}=u", "result": "x=u+\\frac{3}{2}", "steps": [ { "type": "step", "primary": "Add $$\\frac{3}{2}$$ to both sides", "result": "x-\\frac{3}{2}+\\frac{3}{2}=u+\\frac{3}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=u+\\frac{3}{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{4\\left(u+\\frac{3}{2}\\right)}{4u^{2}+27}du" }, { "type": "interim", "title": "Simplify $$\\frac{4\\left(u+\\frac{3}{2}\\right)}{4u^{2}+27}:{\\quad}\\frac{2\\left(2u+3\\right)}{4u^{2}+27}$$", "input": "\\frac{4\\left(u+\\frac{3}{2}\\right)}{4u^{2}+27}", "steps": [ { "type": "interim", "title": "Join $$u+\\frac{3}{2}:{\\quad}\\frac{2u+3}{2}$$", "input": "u+\\frac{3}{2}", "result": "=\\frac{4\\cdot\\:\\frac{2u+3}{2}}{4u^{2}+27}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$u=\\frac{u2}{2}$$", "result": "=\\frac{u\\cdot\\:2}{2}+\\frac{3}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{u\\cdot\\:2+3}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Multiply $$4\\cdot\\:\\frac{u\\cdot\\:2+3}{2}\\::{\\quad}2\\left(2u+3\\right)$$", "input": "4\\cdot\\:\\frac{u\\cdot\\:2+3}{2}", "result": "=\\frac{2\\left(2u+3\\right)}{4u^{2}+27}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(u\\cdot\\:2+3\\right)\\cdot\\:4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2\\left(2u+3\\right)" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{2\\left(2u+3\\right)}{4u^{2}+27}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79CpQmFIaDVorg61uh9xh1NAa4TklNwu59ZvK1TVm+jIQZasjESd2AQANkd7VPJWBpN1pXT08zEQpn0WJ6CFMXCTowm3jYuELCHP4YZPmm8WpbcRWpxEjfgja7oYTobNySAjjcxLNGtqIcwjij5sjyzw2H9uwcr9Of41tVNva+uSLle9ho0biD7NGR0Y47Tyx+bDWr6GMUSMrMlGBgtNEZJP943lRnGl60Ia9GIhzzMJcyQyKTTPwYipg/6j/ntaLQ==" } }, { "type": "step", "result": "=2\\cdot\\:\\int\\:\\frac{2\\left(2u+3\\right)}{4u^{2}+27}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=2\\cdot\\:2\\cdot\\:\\int\\:\\frac{2u+3}{4u^{2}+27}du" }, { "type": "interim", "title": "Expand $$\\frac{2u+3}{4u^{2}+27}:{\\quad}\\frac{2u}{4u^{2}+27}+\\frac{3}{4u^{2}+27}$$", "input": "\\frac{2u+3}{4u^{2}+27}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "result": "=\\frac{2u}{4u^{2}+27}+\\frac{3}{4u^{2}+27}" } ], "meta": { "interimType": "Algebraic Manipulation Expand 1Eq" } }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=2\\cdot\\:2\\left(\\int\\:\\frac{2u}{4u^{2}+27}du+\\int\\:\\frac{3}{4u^{2}+27}du\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{2u}{4u^{2}+27}du=\\frac{1}{4}\\ln\\left|4u^{2}+27\\right|$$", "input": "\\int\\:\\frac{2u}{4u^{2}+27}du", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=2\\cdot\\:\\int\\:\\frac{u}{4u^{2}+27}du" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{u}{4u^{2}+27}du", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$v=4u^{2}+27$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=8u$$", "input": "\\frac{d}{du}\\left(4u^{2}+27\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{du}\\left(4u^{2}\\right)+\\frac{d}{du}\\left(27\\right)" }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(4u^{2}\\right)=8u$$", "input": "\\frac{d}{du}\\left(4u^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{du}\\left(u^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4\\cdot\\:2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmlNBeyYAxp6r2Y83eYMjyaTdaV09PMxEKZ9FieghTFwQFTal+18j5CaTaZWLMNoRqN6Hv6MoTMtvtU0IQwXdn/opg0pDqAhQx5eD06LEE3gvRWRQvqcpnBK6TL4xCiU8A==" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(27\\right)=0$$", "input": "\\frac{d}{du}\\left(27\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr/E8ot2Yrx/vNb/d7rDAFPZGku9zFkxwe1dTH8vycb9TbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51jklTMEwYiOilLrIgbRabD5" } }, { "type": "step", "result": "=8u+0" }, { "type": "step", "primary": "Simplify", "result": "=8u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=8udu$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{1}{8u}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{u}{v}\\cdot\\:\\frac{1}{8u}dv" }, { "type": "interim", "title": "Simplify $$\\frac{u}{v}\\cdot\\:\\frac{1}{8u}:{\\quad}\\frac{1}{8v}$$", "input": "\\frac{u}{v}\\cdot\\:\\frac{1}{8u}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{u\\cdot\\:1}{v\\cdot\\:8u}" }, { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=\\frac{1}{v\\cdot\\:8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{8v}dv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zr9UbG0cs/cN7bqtDmCaCw6q9WBk0dAJfc+XnqLkvyHcSPKmf0CNKWC8kGIkCk3X3KtxwM1n7owdoG2GFz6ksdsfL2GYVRvzzfVlswjxO2YfLzEb1EAHAXUAkBVlbCU4GRLd2VwIqlBNByF6663syQCDi6vmaOkD1HbBCz00khWBTgyp/ddcEl133kNUk+QgA==" } }, { "type": "step", "result": "=2\\cdot\\:\\int\\:\\frac{1}{8v}dv" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=2\\cdot\\:\\frac{1}{8}\\cdot\\:\\int\\:\\frac{1}{v}dv" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{v}dv=\\ln\\left(\\left|v\\right|\\right)$$", "result": "=2\\cdot\\:\\frac{1}{8}\\ln\\left|v\\right|" }, { "type": "step", "primary": "Substitute back $$v=4u^{2}+27$$", "result": "=2\\cdot\\:\\frac{1}{8}\\ln\\left|4u^{2}+27\\right|" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{1}{8}\\ln\\left|4u^{2}+27\\right|:{\\quad}\\frac{1}{4}\\ln\\left|4u^{2}+27\\right|$$", "input": "2\\cdot\\:\\frac{1}{8}\\ln\\left|4u^{2}+27\\right|", "result": "=\\frac{1}{4}\\ln\\left|4u^{2}+27\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{8}\\ln\\left|4u^{2}+27\\right|" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:2}{8}=\\frac{1}{4}$$", "input": "\\frac{1\\cdot\\:2}{8}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{8}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGClXKv6/Z+/k+tIeRZMYPv7Krju+5Z51e/ZZSD3gRHwjB1LXKXwT2OtqKsYdeLWQaeGRLd2VwIqlBNByF6663syS+X17UkJ3twhRGTj59gyRmy3RMq68H7a2lMeRkFYx3vLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{1}{4}\\ln\\left|4u^{2}+27\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvitikmztj1Nlj+LQtdc+hePEycV0cd5WGTKWlXxAMYWF3BFriwl7LUBRTsYITt0UuDv2i9gqKNBiEkMJvG7+cA4lhuGFrgnoTUmHtQOeiDBkG27aK7IHmHf2kgS3hbiRvhr64lnM7s2uGU+hCUmxtlKpN5Aod6Hr1Lp2e/29KhSgU4nw7BGf6PVvnUoBojhEDgP8tbsGVu9S5dbSiDGq+A5I3v07Fp3PH5Pk0b7/nhKTkxRBjrDAHd/Bnj+N8d5jQrQ==" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{3}{4u^{2}+27}du=\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)$$", "input": "\\int\\:\\frac{3}{4u^{2}+27}du", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=3\\cdot\\:\\int\\:\\frac{1}{4u^{2}+27}du" }, { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{1}{4u^{2}+27}du", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=\\frac{3\\sqrt{3}}{2}v$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=27,\\:b=4,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=\\frac{3\\sqrt{3}}{2}\\quad\\Rightarrow\\quad$$substitute $$x=\\frac{3\\sqrt{3}}{2}u$$" }, { "type": "interim", "title": "$$\\frac{du}{dv}=\\frac{3\\sqrt{3}}{2}$$", "input": "\\frac{d}{dv}\\left(\\frac{3\\sqrt{3}}{2}v\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\frac{3\\sqrt{3}}{2}\\frac{dv}{dv}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dv}{dv}=1$$", "result": "=\\frac{3\\sqrt{3}}{2}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{3\\sqrt{3}}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsXWmIOl7hSPupuME1cMrb0iDukztlzx9VrC4+YszdtGk3WldPTzMRCmfRYnoIUxcEhqcfXty6JYd1pWIgQ3Zjgk96XE2/4ak2RL8kN6VIeW/z//r+dXk7h9vxeDCLuZqn45amKCD4X1a8jVcqn6JKjJdMfQ7pd7h1sToWVJaPeq5H63mQLB3jpC3HydXpDGUw==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{3\\sqrt{3}}{2}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}\\cdot\\:\\frac{3\\sqrt{3}}{2}dv" }, { "type": "interim", "title": "Simplify $$\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}\\cdot\\:\\frac{3\\sqrt{3}}{2}:{\\quad}\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}$$", "input": "\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}\\cdot\\:\\frac{3\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}=\\frac{1}{27v^{2}+27}$$", "input": "\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}", "steps": [ { "type": "interim", "title": "$$4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}=27v^{2}$$", "input": "4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}=\\frac{3^{3}v^{2}}{2^{2}}$$", "input": "\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{3\\sqrt{3}}{2}v\\::{\\quad}\\frac{3\\sqrt{3}v}{2}$$", "input": "\\frac{3\\sqrt{3}}{2}v", "result": "=\\left(\\frac{3\\sqrt{3}v}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\sqrt{3}v}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(3\\sqrt{3}v\\right)^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(3\\sqrt{3}v\\right)^{2}=3^{2}\\left(\\sqrt{3}\\right)^{2}v^{2}$$" ], "result": "=\\frac{3^{2}\\left(\\sqrt{3}\\right)^{2}v^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3^{2}\\cdot\\:3v^{2}}{2^{2}}" }, { "type": "interim", "title": "$$3^{2}\\cdot\\:3v^{2}=3^{3}v^{2}$$", "input": "3^{2}\\cdot\\:3v^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$3^{2}\\cdot\\:3=\\:3^{2+1}$$" ], "result": "=3^{2+1}v^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=3^{3}v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vUifPeex/3tg6k39PdaN7o5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjZS5Ux15L2txxMil3n7XewqaV6t7P7d9JflwLBQ2VCZfG4H422xLYkpx3pePR66FtRJpi9Y9E5hYhVqHHTznoAg==" } }, { "type": "step", "result": "=\\frac{3^{3}v^{2}}{2^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7226FmN0dRBsnZkAFqpO2jYBgtMsPjEIiAcn2BQSfrlNV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAluMNxgIcWfbwVDa+pRx3UbPpPiilD70vUmCAUXSjCREpLI3VhNBxbIk2I222da+pCVutje/z1uQEaP/HJOtLbVKyN5OmlaDufYQQY3aczYR10mzzQZlo67PtljK23GTRXA==" } }, { "type": "step", "result": "=4\\cdot\\:\\frac{3^{3}v^{2}}{2^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3^{3}v^{2}\\cdot\\:4}{2^{2}}" }, { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{3^{3}\\cdot\\:2^{2}v^{2}}{2^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$2^{2}$$", "result": "=3^{3}v^{2}" }, { "type": "step", "primary": "$$3^{3}=27$$", "result": "=27v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7u5Erv2YN//c3t9i+ElGPIE5LI5Bbnv0wYH2fTal+m9otOtZYwUjyXhDTsNnn6Elrc5BeeJksUNUaNdP2q+EZ2V5NkzKQgtswLlLi9MgL+goFzkZCH9PCZv7qZcnhAmYOW62N7/PW5ARo/8ck60ttUk5iH+1K7lOcZEHrmEr1lDE=" } }, { "type": "step", "result": "=\\frac{1}{27v^{2}+27}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AlZZZO+ZQCd4FisGK02eyfBAHBObf3Xh1+trRZoy0iHuBWoBFNaI0M/pmL2Kd/on3oZCq59Hq2va8/E5S/sf7wVWfb8eY836P59a0g0AIKYw+3AHonACPPr47ccwUIaSZEt3ZXAiqUE0HIXrrrezJFC5Xe/3aVtCGNJ4TPDjRbfs9D98et4cyldrd3nh25/4n8QXIcx/54r6LomyDaZR3lq1KQafUwRNRjX+toSEtw+wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{3\\sqrt{3}}{2}\\cdot\\:\\frac{1}{27v^{2}+27}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:3\\sqrt{3}}{\\left(27v^{2}+27\\right)\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3\\sqrt{3}}{2\\left(27v^{2}+27\\right)}" }, { "type": "interim", "title": "Factor $$\\left(27v^{2}+27\\right)\\cdot\\:2:{\\quad}54\\left(v^{2}+1\\right)$$", "input": "\\left(27v^{2}+27\\right)\\cdot\\:2", "result": "=\\frac{3\\sqrt{3}}{54\\left(v^{2}+1\\right)}", "steps": [ { "type": "interim", "title": "Factor $$27v^{2}+27:{\\quad}27\\left(v^{2}+1\\right)$$", "input": "27v^{2}+27", "result": "=27\\left(v^{2}+1\\right)\\cdot\\:2", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=27v^{2}+27\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$27$$", "result": "=27\\left(v^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Refine", "result": "=54\\left(v^{2}+1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{\\sqrt{3}}{18\\left(v^{2}+1\\right)}" }, { "type": "interim", "title": "Factor $$18:{\\quad}3^{2}\\cdot\\:2$$", "steps": [ { "type": "step", "primary": "Factor $$18=3^{2}\\cdot\\:2$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{3^{2}\\cdot\\:2\\left(v^{2}+1\\right)}" }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{3}}{2\\cdot\\:3^{2}\\left(v^{2}+1\\right)}:{\\quad}\\frac{1}{2\\cdot\\:3^{\\frac{3}{2}}\\left(v^{2}+1\\right)}$$", "input": "\\frac{\\sqrt{3}}{2\\cdot\\:3^{2}\\left(v^{2}+1\\right)}", "result": "=\\frac{1}{2\\cdot\\:3^{\\frac{3}{2}}\\left(v^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{3}=3^{\\frac{1}{2}}$$" ], "result": "=\\frac{3^{\\frac{1}{2}}}{3^{2}\\cdot\\:2\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{3^{\\frac{1}{2}}}{3^{2}}=\\frac{1}{3^{2-\\frac{1}{2}}}$$" ], "result": "=\\frac{1}{2\\cdot\\:3^{-\\frac{1}{2}+2}\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=\\frac{1}{2\\cdot\\:3^{\\frac{3}{2}}\\left(v^{2}+1\\right)}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlrPaNU/DZnBhshfbTfWKQq5S7tpEHMtunxkoVHXtbRUJEF3sabxg7oc35wZwdgNCs72GhpsGZb8uajHPNlSOnWVuqXiNB9za+2c/S23QICkRFxJViUpAI13eo1l+oTyNSEwLIR/qEc/t9wqmVfLrA3wt9LEn7QCBUukJKctfSJK33L7StcEuV/XY4CWwV/jWO3fQqaNbsq8jrIRXzhy/5uTiZ2DGuIXfh4FNJKtdzQkEmcRvx50wuMYcUqA2dQQYQ==" } }, { "type": "interim", "title": "$$3^{\\frac{3}{2}}=3\\sqrt{3}$$", "input": "3^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "$$3^{\\frac{3}{2}}=3^{1+\\frac{1}{2}}$$", "result": "=3^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=3^{1}\\cdot\\:3^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=3\\sqrt{3}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{1}{2\\cdot\\:3\\sqrt{3}\\left(v^{2}+1\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}dv" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=3\\cdot\\:\\int\\:\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}dv" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=3\\cdot\\:\\frac{1}{6\\sqrt{3}}\\cdot\\:\\int\\:\\frac{1}{v^{2}+1}dv" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{v^{2}+1}dv=\\arctan\\left(v\\right)$$", "result": "=3\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(v\\right)" }, { "type": "step", "primary": "Substitute back $$v=\\frac{2}{3\\sqrt{3}}u$$", "result": "=3\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)" }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right):{\\quad}\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)$$", "input": "3\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)", "result": "=\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:3}{6\\sqrt{3}}=\\frac{1}{2\\sqrt{3}}$$", "input": "\\frac{1\\cdot\\:3}{6\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{6\\sqrt{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{1}{2\\sqrt{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCrOxLbJpv+10z+fbRBAZ+lctOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOPDrE+QzdkdlVkDqbC3eAMX/P/+v51eTuH2/F4MIu5mqGk7eu+zvKmnz29/GR90k8ANrB4/rfbJqurqKv1tCxG1qew46gc3e2k9WpyQPp1UJzru4mJyAykvNbli74dD+NA==" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA14aUXjHeQh+UlhVlRPJgOyxssVsfkX7bGZ3jlM8cWJc53ptS43jMzqqtjiypyrTyi061ljBSPJeENOw2efoSWtIanH17cuiWHdaViIEN2Y48OsT5DN2R2VWQOpsLd4Axbl0leXxZ+4tr1eU3oO4fSeyqJp7sdc7wD8XlEh6l7Gl7lvKtANBUJdQPS8f9+853HKF3u2OIb4bFA3EO8aRlSVnO24yvsGViR8V5sSDsWBoBdAacHgGZluUV+LE6uW7iLl0leXxZ+4tr1eU3oO4fSeyqJp7sdc7wD8XlEh6l7Gl4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=2\\cdot\\:2\\left(\\frac{1}{4}\\ln\\left|4u^{2}+27\\right|+\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-\\frac{3}{2}$$", "result": "=2\\cdot\\:2\\left(\\frac{1}{4}\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|+\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\left(\\frac{1}{4}\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|+\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)\\right):{\\quad}\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$", "input": "2\\cdot\\:2\\left(\\frac{1}{4}\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|+\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)\\right)", "result": "=\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{1}{4}\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|=\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}$$", "input": "\\frac{1}{4}\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|}{4}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|=\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|$$", "result": "=\\frac{\\ln\\left|4\\left(x-\\frac{3}{2}\\right)^{2}+27\\right|}{4}" }, { "type": "interim", "title": "$$\\left(x-\\frac{3}{2}\\right)^{2}=x^{2}-3x+\\frac{9}{4}$$", "input": "\\left(x-\\frac{3}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=\\frac{3}{2}$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}-2x\\frac{3}{2}+\\left(\\frac{3}{2}\\right)^{2}" }, { "type": "interim", "title": "Simplify $$x^{2}-2x\\frac{3}{2}+\\left(\\frac{3}{2}\\right)^{2}:{\\quad}x^{2}-3x+\\frac{9}{4}$$", "input": "x^{2}-2x\\frac{3}{2}+\\left(\\frac{3}{2}\\right)^{2}", "result": "=x^{2}-3x+\\frac{9}{4}", "steps": [ { "type": "interim", "title": "$$2x\\frac{3}{2}=3x$$", "input": "2x\\frac{3}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\cdot\\:2}{2}x" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=x\\cdot\\:3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vERBoEQkIKdGT/H8fwqIVQCWKUbvV6WK3fDUgFtg3Q/ikalcfvsjMVQXpWJ/icqAgabmpCgj1TW/wXUnzpvONfa9oGnHhqTo0qy+mUerqD/PjltOhgG2v9g02eWQv40e" } }, { "type": "interim", "title": "$$\\left(\\frac{3}{2}\\right)^{2}=\\frac{9}{4}$$", "input": "\\left(\\frac{3}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{3^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=\\frac{9}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7le+b864qvC+j2iBYaWnVJo5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjl+AzEfHKc8lPiV91zCrgMlaiLgjmyMQYlA0xnylLMSczG2XZl3KNwToSWv+5aI5+AV7fhkphgvLSgCLTYGcO+g==" } }, { "type": "step", "result": "=x^{2}-3x+\\frac{9}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72tEf2RJe5HGGOpqSqzIvxkBgnzAZkUy+vMtgHHTjQTCrju+5Z51e/ZZSD3gRHwjBJ+E1SrTm3lcUeWTedLJirHtdqLmMH5jlexKmZU8C6eNymqb7o0A6pfQwKBwKm1Culne5x1IT8qhj6IsUbH8IYFAvefWPOnon7Yh0y/08zoa3eZAz+5j4a254NnOVeBY5" } }, { "type": "step", "result": "=\\frac{\\ln\\left|4\\left(x^{2}-3x+\\frac{9}{4}\\right)+27\\right|}{4}" }, { "type": "interim", "title": "Expand $$4\\left(x^{2}-3x+\\frac{9}{4}\\right)+27:{\\quad}4x^{2}-12x+36$$", "input": "4\\left(x^{2}-3x+\\frac{9}{4}\\right)+27", "result": "=\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}", "steps": [ { "type": "interim", "title": "Expand $$4\\left(x^{2}-3x+\\frac{9}{4}\\right):{\\quad}4x^{2}-12x+9$$", "input": "4\\left(x^{2}-3x+\\frac{9}{4}\\right)", "result": "=4x^{2}-12x+9+27", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=4x^{2}+4\\left(-3x\\right)+4\\cdot\\:\\frac{9}{4}", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=4x^{2}-4\\cdot\\:3x+4\\cdot\\:\\frac{9}{4}" }, { "type": "interim", "title": "Simplify $$4x^{2}-4\\cdot\\:3x+4\\cdot\\:\\frac{9}{4}:{\\quad}4x^{2}-12x+9$$", "input": "4x^{2}-4\\cdot\\:3x+4\\cdot\\:\\frac{9}{4}", "result": "=4x^{2}-12x+9", "steps": [ { "type": "interim", "title": "$$4\\cdot\\:3x=12x$$", "input": "4\\cdot\\:3x", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3=12$$", "result": "=12x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76DaBQ/clmEaZdiossBc21iAn9lkDfZkicUGkO3EF+Io7TJgXBzmRSM2ZmvJL/c8RjcgfmBx2/t6xjA5NWoZy2e2DPiP6gFy7zfnQoUQQPbc=" } }, { "type": "interim", "title": "$$4\\cdot\\:\\frac{9}{4}=9$$", "input": "4\\cdot\\:\\frac{9}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{9\\cdot\\:4}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=9" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV85zTwmsUFZ1CMMGKJayQvmmrju+5Z51e/ZZSD3gRHwjBZO3KH1ZhYMuwWllqzjrZFLCmhlNavddal7Sw5Ic5ePuZM9n2H4Sv3+O7ae9hylwy" } }, { "type": "step", "result": "=4x^{2}-12x+9" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s788BZbTMqZjnFM6BxFI2l6v/txaKTQvEKi0mouRAxr3PNGoPE9TME3q+OPmgkv2RQ1bAlWcd9rNvkD2EQTPZQFc6CR3WRNWA4W+YHteKP7kve1A9ekahUIWtvNWrN4v94UuaNk7F0/BxKwZjtEpZ4EMRXW34FSdgQMpa0Zjbx9S4=" } }, { "type": "step", "primary": "Add the numbers: $$9+27=36$$", "result": "=4x^{2}-12x+36" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s788BZbTMqZjnFM6BxFI2l6rJ0G/JZyzUqxdRLqjCkKgfehkKrn0era9rz8TlL+x/vEn4R/y9LnvrzR5AgvkgmpkoFufqF5YUQNGwkbtWMNB1N5Aod6Hr1Lp2e/29KhSgUEl/FdTFrncvVVSjz/L1NkmM8vBFnLPP/DDlJDcz+rFho+YhnyF7YWVsyN3Fy4xd9" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/+LVvf+dQpmEy47zPJ1c6aLhUIGeTdJDe2iD4fpa+nx1iIckGWd4xQc7rAPrxSN5ien95tRr14EweFxWyXZp+gOfOVs9mPIqDLV5QIWwt3kTXc5rzUBtgav0y+wkj/ZZI2i7Bz8JQRgzWNQsrbT4IMc9KPYe9rTXsRxjv5TXOEn/P/+v51eTuH2/F4MIu5mq7onu565T+ddaWcJRu3dS17clnE5a9CdcjbsdXtNhvUnICTfpIkAhf8BlAjk57zIsZkRCXiUVi+4jM6UAQCngm45BUD9vUXXjaJBHfuA7DY7jpLwbQQo5f7i/SgvFyYCYuxq/qgrA5qWTcQ1KY33A1Q==" } }, { "type": "interim", "title": "$$\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)=\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}$$", "input": "\\frac{1}{2\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)}{2\\sqrt{3}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)=\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)$$", "result": "=\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)}{2\\sqrt{3}}" }, { "type": "interim", "title": "Expand $$\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right):{\\quad}\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}$$", "input": "\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)", "result": "=\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=\\frac{2}{3\\sqrt{3}},\\:b=x,\\:c=\\frac{3}{2}$$" ], "result": "=\\frac{2}{3\\sqrt{3}}x-\\frac{2}{3\\sqrt{3}}\\cdot\\:\\frac{3}{2}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=\\frac{2}{3\\sqrt{3}}x-\\frac{3}{2}\\cdot\\:\\frac{2}{3\\sqrt{3}}" }, { "type": "interim", "title": "$$\\frac{3}{2}\\cdot\\:\\frac{2}{3\\sqrt{3}}=\\frac{1}{\\sqrt{3}}$$", "input": "\\frac{3}{2}\\cdot\\:\\frac{2}{3\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Cross-cancel: $$2$$", "result": "=\\frac{3}{3\\sqrt{3}}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=\\frac{1}{\\sqrt{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hbqZttgXqBCr9XmUErpy5YM8PijhZswI5GTIAEVyDfa/mnJGH0ehb0t8I5YbKMNHzRqDxPUzBN6vjj5oJL9kUIhMOhuE/kRSNmX2Q6Ad5DIu7p1yw0yG3n/qbNxp9TjoDht/CJXqIwsnmwLWcvt05ljVMn1G/urvbkDD6XnKpRxTl+3uPJcRHv8hz8GwIc0l/oolhrgn+BnmZVGuPaxjFdNFiEsD4HzUtKtBhF1mmtU=" } }, { "type": "step", "result": "=\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FASQQv+6SyVcKr3SljvYOtZvtJeh37VUgkLUXmbijVvBhTusSzdkh4TMNmZM9WtZCUCWbkwGOY7PqKo3U/JLJZhEyNSYUmHqHNOStzOxNtst8nQpe3lYiNMCyXAK3socEUF/Rf7rCF/pDoAV8M4HwGRLd2VwIqlBNByF6663syRU6H0nS++8kDqP632fuVHPuq88PmTTiFdgD4eGshPPIGoeyp/txmr71cljrGYgHa5LiF4Mzbt6swySWFVTee4m" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=2\\cdot\\:2\\left(\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}+\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\left(\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}+\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=4,\\:b=\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4},\\:c=\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}$$" ], "result": "=4\\cdot\\:\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}+4\\cdot\\:\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$4\\cdot\\:\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}+4\\cdot\\:\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}:{\\quad}\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}$$", "input": "4\\cdot\\:\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}+4\\cdot\\:\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}", "result": "=\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}", "steps": [ { "type": "interim", "title": "$$4\\cdot\\:\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}=\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "4\\cdot\\:\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\ln\\left|4x^{2}-12x+36\\right|\\cdot\\:4}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\ln\\left|4x^{2}-12x+36\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV85K0Pcadwd5bclWgOfL6qT83/ZgZoreBLMQFmdfklhTiv4+BRdpwkpVq6ZkPZm7BuHCQoYlYQ8U+Tfyx0kyzI8i1mdrueH9IYWDWhLEbHr9UuBVHldoZBMfSza7e1beHdccnbR3EVU1oCuGutPo/e8IRdAMgAa6z4abUn0SeiuaJkrQ9xp3B3ltyVaA58vqpPzf9mBmit4EsxAWZ1+SWFOKVca0XeTu9bHrgQ31EgAg2d8n2bFKTNJGuaCiUCq02Ta9XM7WEZOM4xt3TBOoolAo=" } }, { "type": "interim", "title": "$$4\\cdot\\:\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}=\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}$$", "input": "4\\cdot\\:\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{2\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\cdot\\:4}{2\\sqrt{3}}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{3}{x^{2}-3x+9}dx=\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$", "input": "\\int\\:\\frac{3}{x^{2}-3x+9}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=3\\cdot\\:\\int\\:\\frac{1}{x^{2}-3x+9}dx" }, { "type": "interim", "title": "Complete the square $$x^{2}-3x+9:{\\quad}\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}$$", "input": "x^{2}-3x+9", "steps": [ { "type": "step", "primary": "Write $$x^{2}-3x+9\\:$$in the form: $$x^2+2bx+b^2$$" }, { "type": "interim", "title": "$$2b=-3{\\quad:\\quad}b=-\\frac{3}{2}$$", "input": "2b=-3", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2b=-3", "result": "b=-\\frac{3}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2b}{2}=\\frac{-3}{2}" }, { "type": "step", "primary": "Simplify", "result": "b=-\\frac{3}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VUsluyXsY1Ps8efqNyHMAJN1pXT08zEQpn0WJ6CFMXBgNJ67EOD8lvNAD3ifVxEo6ChE8r/ZN/+ka9wpb6F5VgJjvXUNhu45aE9syEoEnEp7CO2C/Y/8sYuRjRA2n7M+z27xpiicCLlej+r3Kqpg6rOnDLD0jssdVCDGYRPtoVQRv99Fvop71hs19i20lNqnlYELdzcGYO9oywqqeNLwu+4cxhrmNyLX7xBQkB3NzBMOaqlvSGagjUxH5yDFsFwS3vIvZmQvOu29BpwLovwPXcpXnCSXRU+Ii63Z3TnPUTDWqq4EX5rIYM9pz1jWVJU/8yB/Byh4uWS4fw659PzN/S+Qy/bIAMBOiqTKlPRoUSwD2vk/DYOs8QR1k++gTUdg2PCKH5nJgiSZ4BgekfLbG9IdGwGoj7rR26AEKfEFIcFqtoU1FAFGQMOcL+h4fgMTGo/fecg+jRtJU+qFH+NxRAwKAu7Z3tN/RgLPjlzyRw8i5OoNNprqQV2UgQUGESqFlvjhqQs5XJJkHv2GQUE6glcrAOsROfiVgP+wtWyh0afthvwKEoR3ksHNHbCkrb6kcwDIfxbr/Pq2vWjsXBdpmP+Gc7uNDn5rFMiDyv69niKC8VZfRwvJI6wrKTATUX8dU9cwwVwt4DorvBqyqIystnWNWjc0q0gZIiqs/HmA5vE=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-\\frac{3}{2}\\right)^{2}\\:$$", "result": "=x^{2}-3x+9+\\left(-\\frac{3}{2}\\right)^{2}-\\left(-\\frac{3}{2}\\right)^{2}" }, { "type": "step", "primary": "$$x^2+2bx+b^2=\\left(x+b\\right)^2$$", "secondary": [ "$$x^{2}-3x+\\left(-\\frac{3}{2}\\right)^{2}=\\left(x-\\frac{3}{2}\\right)^{2}$$", "Complete the square" ], "result": "=\\left(x-\\frac{3}{2}\\right)^{2}+9-\\left(-\\frac{3}{2}\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "=\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "=3\\cdot\\:\\int\\:\\frac{1}{\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{\\left(x-\\frac{3}{2}\\right)^{2}+\\frac{27}{4}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=x-\\frac{3}{2}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x-\\frac{3}{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(\\frac{3}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{3}{2}\\right)=0$$", "input": "\\frac{d}{dx}\\left(\\frac{3}{2}\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnHXdJ1FY75umoXux1bL9K944OmnsvrgbNkOIEFbFW8WYAIXwDG7/aP+CNC9gUVzaHNS9SX5M3gDB/Er/MAH1V+5QV7agSZLIzF7D9vX0CHvUmusDuyFta1vRswttwqLuLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=1du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{2}+\\frac{27}{4}}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{4}{4u^{2}+27}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70wyau0cnlcL4sWSHGzSBFNAa4TklNwu59ZvK1TVm+jIQZasjESd2AQANkd7VPJWBpN1pXT08zEQpn0WJ6CFMXCTowm3jYuELCHP4YZPmm8WpbcRWpxEjfgja7oYTobNySAjjcxLNGtqIcwjij5sjywf8qzt92tPBHY3D9jGsbzzZEt3ZXAiqUE0HIXrrrezJAIOLq+Zo6QPUdsELPTSSFYFODKn911wSXXfeQ1ST5CA" } }, { "type": "step", "result": "=3\\cdot\\:\\int\\:\\frac{4}{4u^{2}+27}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=3\\cdot\\:4\\cdot\\:\\int\\:\\frac{1}{4u^{2}+27}du" }, { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{1}{4u^{2}+27}du", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=\\frac{3\\sqrt{3}}{2}v$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=27,\\:b=4,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=\\frac{3\\sqrt{3}}{2}\\quad\\Rightarrow\\quad$$substitute $$x=\\frac{3\\sqrt{3}}{2}u$$" }, { "type": "interim", "title": "$$\\frac{du}{dv}=\\frac{3\\sqrt{3}}{2}$$", "input": "\\frac{d}{dv}\\left(\\frac{3\\sqrt{3}}{2}v\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\frac{3\\sqrt{3}}{2}\\frac{dv}{dv}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dv}{dv}=1$$", "result": "=\\frac{3\\sqrt{3}}{2}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{3\\sqrt{3}}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsXWmIOl7hSPupuME1cMrb0iDukztlzx9VrC4+YszdtGk3WldPTzMRCmfRYnoIUxcEhqcfXty6JYd1pWIgQ3Zjgk96XE2/4ak2RL8kN6VIeW/z//r+dXk7h9vxeDCLuZqn45amKCD4X1a8jVcqn6JKjJdMfQ7pd7h1sToWVJaPeq5H63mQLB3jpC3HydXpDGUw==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{3\\sqrt{3}}{2}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}\\cdot\\:\\frac{3\\sqrt{3}}{2}dv" }, { "type": "interim", "title": "Simplify $$\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}\\cdot\\:\\frac{3\\sqrt{3}}{2}:{\\quad}\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}$$", "input": "\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}\\cdot\\:\\frac{3\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}=\\frac{1}{27v^{2}+27}$$", "input": "\\frac{1}{4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}+27}", "steps": [ { "type": "interim", "title": "$$4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}=27v^{2}$$", "input": "4\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}=\\frac{3^{3}v^{2}}{2^{2}}$$", "input": "\\left(\\frac{3\\sqrt{3}}{2}v\\right)^{2}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{3\\sqrt{3}}{2}v\\::{\\quad}\\frac{3\\sqrt{3}v}{2}$$", "input": "\\frac{3\\sqrt{3}}{2}v", "result": "=\\left(\\frac{3\\sqrt{3}v}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\sqrt{3}v}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(3\\sqrt{3}v\\right)^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(3\\sqrt{3}v\\right)^{2}=3^{2}\\left(\\sqrt{3}\\right)^{2}v^{2}$$" ], "result": "=\\frac{3^{2}\\left(\\sqrt{3}\\right)^{2}v^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3^{2}\\cdot\\:3v^{2}}{2^{2}}" }, { "type": "interim", "title": "$$3^{2}\\cdot\\:3v^{2}=3^{3}v^{2}$$", "input": "3^{2}\\cdot\\:3v^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$3^{2}\\cdot\\:3=\\:3^{2+1}$$" ], "result": "=3^{2+1}v^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=3^{3}v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vUifPeex/3tg6k39PdaN7o5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjZS5Ux15L2txxMil3n7XewqaV6t7P7d9JflwLBQ2VCZfG4H422xLYkpx3pePR66FtRJpi9Y9E5hYhVqHHTznoAg==" } }, { "type": "step", "result": "=\\frac{3^{3}v^{2}}{2^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7226FmN0dRBsnZkAFqpO2jYBgtMsPjEIiAcn2BQSfrlNV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAluMNxgIcWfbwVDa+pRx3UbPpPiilD70vUmCAUXSjCREpLI3VhNBxbIk2I222da+pCVutje/z1uQEaP/HJOtLbVKyN5OmlaDufYQQY3aczYR10mzzQZlo67PtljK23GTRXA==" } }, { "type": "step", "result": "=4\\cdot\\:\\frac{3^{3}v^{2}}{2^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3^{3}v^{2}\\cdot\\:4}{2^{2}}" }, { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{3^{3}\\cdot\\:2^{2}v^{2}}{2^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$2^{2}$$", "result": "=3^{3}v^{2}" }, { "type": "step", "primary": "$$3^{3}=27$$", "result": "=27v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7u5Erv2YN//c3t9i+ElGPIE5LI5Bbnv0wYH2fTal+m9otOtZYwUjyXhDTsNnn6Elrc5BeeJksUNUaNdP2q+EZ2V5NkzKQgtswLlLi9MgL+goFzkZCH9PCZv7qZcnhAmYOW62N7/PW5ARo/8ck60ttUk5iH+1K7lOcZEHrmEr1lDE=" } }, { "type": "step", "result": "=\\frac{1}{27v^{2}+27}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AlZZZO+ZQCd4FisGK02eyfBAHBObf3Xh1+trRZoy0iHuBWoBFNaI0M/pmL2Kd/on3oZCq59Hq2va8/E5S/sf7wVWfb8eY836P59a0g0AIKYw+3AHonACPPr47ccwUIaSZEt3ZXAiqUE0HIXrrrezJFC5Xe/3aVtCGNJ4TPDjRbfs9D98et4cyldrd3nh25/4n8QXIcx/54r6LomyDaZR3lq1KQafUwRNRjX+toSEtw+wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{3\\sqrt{3}}{2}\\cdot\\:\\frac{1}{27v^{2}+27}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:3\\sqrt{3}}{\\left(27v^{2}+27\\right)\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3\\sqrt{3}}{2\\left(27v^{2}+27\\right)}" }, { "type": "interim", "title": "Factor $$\\left(27v^{2}+27\\right)\\cdot\\:2:{\\quad}54\\left(v^{2}+1\\right)$$", "input": "\\left(27v^{2}+27\\right)\\cdot\\:2", "result": "=\\frac{3\\sqrt{3}}{54\\left(v^{2}+1\\right)}", "steps": [ { "type": "interim", "title": "Factor $$27v^{2}+27:{\\quad}27\\left(v^{2}+1\\right)$$", "input": "27v^{2}+27", "result": "=27\\left(v^{2}+1\\right)\\cdot\\:2", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=27v^{2}+27\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$27$$", "result": "=27\\left(v^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Refine", "result": "=54\\left(v^{2}+1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{\\sqrt{3}}{18\\left(v^{2}+1\\right)}" }, { "type": "interim", "title": "Factor $$18:{\\quad}3^{2}\\cdot\\:2$$", "steps": [ { "type": "step", "primary": "Factor $$18=3^{2}\\cdot\\:2$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{3^{2}\\cdot\\:2\\left(v^{2}+1\\right)}" }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{3}}{2\\cdot\\:3^{2}\\left(v^{2}+1\\right)}:{\\quad}\\frac{1}{2\\cdot\\:3^{\\frac{3}{2}}\\left(v^{2}+1\\right)}$$", "input": "\\frac{\\sqrt{3}}{2\\cdot\\:3^{2}\\left(v^{2}+1\\right)}", "result": "=\\frac{1}{2\\cdot\\:3^{\\frac{3}{2}}\\left(v^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{3}=3^{\\frac{1}{2}}$$" ], "result": "=\\frac{3^{\\frac{1}{2}}}{3^{2}\\cdot\\:2\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{3^{\\frac{1}{2}}}{3^{2}}=\\frac{1}{3^{2-\\frac{1}{2}}}$$" ], "result": "=\\frac{1}{2\\cdot\\:3^{-\\frac{1}{2}+2}\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=\\frac{1}{2\\cdot\\:3^{\\frac{3}{2}}\\left(v^{2}+1\\right)}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlrPaNU/DZnBhshfbTfWKQq5S7tpEHMtunxkoVHXtbRUJEF3sabxg7oc35wZwdgNCs72GhpsGZb8uajHPNlSOnWVuqXiNB9za+2c/S23QICkRFxJViUpAI13eo1l+oTyNSEwLIR/qEc/t9wqmVfLrA3wt9LEn7QCBUukJKctfSJK33L7StcEuV/XY4CWwV/jWO3fQqaNbsq8jrIRXzhy/5uTiZ2DGuIXfh4FNJKtdzQkEmcRvx50wuMYcUqA2dQQYQ==" } }, { "type": "interim", "title": "$$3^{\\frac{3}{2}}=3\\sqrt{3}$$", "input": "3^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "$$3^{\\frac{3}{2}}=3^{1+\\frac{1}{2}}$$", "result": "=3^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=3^{1}\\cdot\\:3^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=3\\sqrt{3}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{1}{2\\cdot\\:3\\sqrt{3}\\left(v^{2}+1\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}dv" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=3\\cdot\\:4\\cdot\\:\\int\\:\\frac{1}{6\\sqrt{3}\\left(v^{2}+1\\right)}dv" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\cdot\\:\\int\\:\\frac{1}{v^{2}+1}dv" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{v^{2}+1}dv=\\arctan\\left(v\\right)$$", "result": "=3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(v\\right)" }, { "type": "interim", "title": "Substitute back", "input": "3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(v\\right)", "result": "=3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)", "steps": [ { "type": "step", "primary": "Substitute back $$v=\\frac{2}{3\\sqrt{3}}u$$", "result": "=3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}u\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-\\frac{3}{2}$$", "result": "=3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right):{\\quad}\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$", "input": "3\\cdot\\:4\\cdot\\:\\frac{1}{6\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)", "result": "=\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3\\cdot\\:4\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)}{6\\sqrt{3}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3\\cdot\\:4=12$$", "result": "=\\frac{12\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)}{6\\sqrt{3}}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{6}=2$$", "result": "=\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)\\right)}{\\sqrt{3}}" }, { "type": "interim", "title": "Expand $$\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right):{\\quad}\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}$$", "input": "\\frac{2}{3\\sqrt{3}}\\left(x-\\frac{3}{2}\\right)", "result": "=\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=\\frac{2}{3\\sqrt{3}},\\:b=x,\\:c=\\frac{3}{2}$$" ], "result": "=\\frac{2}{3\\sqrt{3}}x-\\frac{2}{3\\sqrt{3}}\\cdot\\:\\frac{3}{2}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=\\frac{2}{3\\sqrt{3}}x-\\frac{3}{2}\\cdot\\:\\frac{2}{3\\sqrt{3}}" }, { "type": "interim", "title": "$$\\frac{3}{2}\\cdot\\:\\frac{2}{3\\sqrt{3}}=\\frac{1}{\\sqrt{3}}$$", "input": "\\frac{3}{2}\\cdot\\:\\frac{2}{3\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Cross-cancel: $$2$$", "result": "=\\frac{3}{3\\sqrt{3}}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=\\frac{1}{\\sqrt{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hbqZttgXqBCr9XmUErpy5YM8PijhZswI5GTIAEVyDfa/mnJGH0ehb0t8I5YbKMNHzRqDxPUzBN6vjj5oJL9kUIhMOhuE/kRSNmX2Q6Ad5DIu7p1yw0yG3n/qbNxp9TjoDht/CJXqIwsnmwLWcvt05ljVMn1G/urvbkDD6XnKpRxTl+3uPJcRHv8hz8GwIc0l/oolhrgn+BnmZVGuPaxjFdNFiEsD4HzUtKtBhF1mmtU=" } }, { "type": "step", "result": "=\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FASQQv+6SyVcKr3SljvYOtZvtJeh37VUgkLUXmbijVvBhTusSzdkh4TMNmZM9WtZCUCWbkwGOY7PqKo3U/JLJZhEyNSYUmHqHNOStzOxNtst8nQpe3lYiNMCyXAK3socEUF/Rf7rCF/pDoAV8M4HwGRLd2VwIqlBNByF6663syRU6H0nS++8kDqP632fuVHPuq88PmTTiFdgD4eGshPPIGoeyp/txmr71cljrGYgHa5LiF4Mzbt6swySWFVTee4m" } }, { "type": "step", "result": "=\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+CNSC7QMg7u0JdlExWav9RA0Ce/XSHyd+3rlzH43vl8PFdj1FX7Szs35xCPBOzbiuq88PmTTiFdgD4eGshPPIGoeyp/txmr71cljrGYgHa7CnjwC1YTN1VNRbgJ2Hthtq47vuWedXv2WUg94ER8Iwbufkmzwib9pqPA2eECfrNcPFdj1FX7Szs35xCPBOzbiuq88PmTTiFdgD4eGshPPICVNM1GBKbKRQhvzY/577NpK7m7SFitzBG0aFpFG4Ljv72wZm7kDUxdE6YSmfEbr2tZxM3jDj0h4CKI+6lSMXrtU6IVTlpkIkkM1mHOnb5Bm5Fi+eAdoN4sTW6tO34zg82o8m5q4sp0ihsHFgP7/rGhBrePp16GYm3kEcLiFhNSDNPlzqtaA8vQYm0VIg4mLa1/NIQ31PpeZavrJ86BbE/0=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{1}{81}\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{81}\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right):{\\quad}-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "\\frac{1}{81}\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)", "result": "=-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)}{81}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)=-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$", "input": "1\\cdot\\:\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)=\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)$$", "result": "=\\left(-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=\\frac{-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{81}" }, { "type": "interim", "title": "Join $$-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right):{\\quad}-\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "result": "=\\frac{-\\ln\\left|4x^{2}-12x+36\\right|}{81}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\::{\\quad}\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}$$", "input": "\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "result": "=-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Convert element to fraction: $$\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)=\\frac{\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\cdot\\:\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)\\sqrt{3}}{\\sqrt{3}}$$", "result": "=-\\frac{\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)\\sqrt{3}}{\\sqrt{3}}+\\frac{2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)\\sqrt{3}+2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)}{\\sqrt{3}}" }, { "type": "interim", "title": "Expand $$-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)\\sqrt{3}+2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right):{\\quad}-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "-\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)\\sqrt{3}+2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "result": "=\\frac{-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|}{\\sqrt{3}}", "steps": [ { "type": "step", "result": "=-\\sqrt{3}\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)+2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)" }, { "type": "interim", "title": "Expand $$-\\sqrt{3}\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right):{\\quad}-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|-2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$", "input": "-\\sqrt{3}\\left(\\ln\\left|4x^{2}-12x+36\\right|+\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\right)", "result": "=-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|-2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)+2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-\\sqrt{3},\\:b=\\ln\\left|4x^{2}-12x+36\\right|,\\:c=\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$" ], "result": "=-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|+\\left(-\\sqrt{3}\\right)\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|-\\sqrt{3}\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)" }, { "type": "interim", "title": "$$\\sqrt{3}\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)=2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)$$", "input": "\\sqrt{3}\\frac{2}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2\\sqrt{3}}{\\sqrt{3}}\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)\\cdot\\:2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|-2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Add similar elements: $$-2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)+2\\arctan\\left(\\frac{2}{3\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)=0$$", "result": "=-\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7exyiW8SilbEMo3dJEmGczhluxm4zGNwBwmQMCxW2TLUjGjKlB/TfP1l9Ic0CTdGVMo9EWgpbOzeU5UBA1FLT37IIuEkbZo1ycBe6w14wVZ6MN+JAd688fEnZoJAu2nSiIVMK7il4YPDtm4eEAis2CVG4m7ZZo1PQogEUPoKM1amH4Xg57joZhGkaQtuxn+FNOs0bPS1R/OIjCXEqFba617sch8vNbEWzsaAq0qB2BSctOtZYwUjyXhDTsNnn6ElrJZ7s7lv1pf9Tkz9FuXr4VaNAv+k4JT8qRfcbEQRSUZB3yfZsUpM0ka5oKJQKrTZNzi65DrXCDIulsviZu4HM5kDLjX7DcYtB54q3geRNejHsq6UQm2NzdKeumvccYwrhbmxhhjSQ9p09Yue/wWHY0uFdhyt0CwDcFTQoba3c+Cj/fJWiWjfhW8ij2ErhbpVjOA2WPK1yC2ZtmueFJvj7fNqRANYShB3EPsoura5dJHt1tIJlmBRmbqjAoEaa1b8XtMfjRH2m0P2jj3mFc/LynbGyxWx+RftsZneOUzxxYlytMYNODhssuhR+/f/emVzeS85ZTGQFfxkB6fHRDuC+4oi58JLcDV9OjSe4/+Jvhak=" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\sqrt{3}\\ln\\left|4x^{2}-12x+36\\right|}{\\sqrt{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=-\\ln\\left|4x^{2}-12x+36\\right|" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\ln\\left|4x^{2}-12x+36\\right|}{81}" }, { "type": "step", "result": "=-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=81\\left(\\frac{1}{27}\\ln\\left|x\\right|-\\frac{1}{81}\\ln\\left|x+3\\right|-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|\\right)" }, { "type": "interim", "title": "Simplify $$81\\left(\\frac{1}{27}\\ln\\left|x\\right|-\\frac{1}{81}\\ln\\left|x+3\\right|-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|\\right):{\\quad}3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "81\\left(\\frac{1}{27}\\ln\\left|x\\right|-\\frac{1}{81}\\ln\\left|x+3\\right|-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|\\right)", "result": "=3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=81\\cdot\\:\\frac{1}{27}\\ln\\left|x\\right|+81\\left(-\\frac{1}{81}\\ln\\left|x+3\\right|\\right)+81\\left(-\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|\\right)", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=81\\cdot\\:\\frac{1}{27}\\ln\\left|x\\right|-81\\cdot\\:\\frac{1}{81}\\ln\\left|x+3\\right|-81\\cdot\\:\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|" }, { "type": "interim", "title": "Simplify $$81\\cdot\\:\\frac{1}{27}\\ln\\left|x\\right|-81\\cdot\\:\\frac{1}{81}\\ln\\left|x+3\\right|-81\\cdot\\:\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|:{\\quad}3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "81\\cdot\\:\\frac{1}{27}\\ln\\left|x\\right|-81\\cdot\\:\\frac{1}{81}\\ln\\left|x+3\\right|-81\\cdot\\:\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|", "result": "=3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|", "steps": [ { "type": "interim", "title": "$$81\\cdot\\:\\frac{1}{27}\\ln\\left|x\\right|=3\\ln\\left|x\\right|$$", "input": "81\\cdot\\:\\frac{1}{27}\\ln\\left|x\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:81}{27}\\ln\\left|x\\right|" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:81}{27}=3$$", "input": "\\frac{1\\cdot\\:81}{27}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:81=81$$", "result": "=\\frac{81}{27}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{81}{27}=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCnN1B809SwXv7kCLxLFXgut1g99dC9fj9sg0EHzBIRDRdf+ZSwzoMmvk4a7GmXcmfCF6hB5GTJ4ecV/JJbovYjZRDksuMfS6hQYsN09MJ8oYJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=3\\ln\\left|x\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OxzBHNC/gxp2+CR/BZ0VwVqFk3cQbpSpHJkRc+K8EVPpT2UOtAFGlX54NzR/8azMICf2WQN9mSJxQaQ7cQX4it+BebMx/La21eIBLGDN8exOzXvUyGSCVtxRSdm2Ytlo9HDuu+zd4MhJLzL6xy/VYXiXj0Ip1doXRPIJCQFOu2QflWgr/LzjX+HfvBokirEi2Lfoz23qxYC3VenVCOhOL3IhG7K/PxQy10OqOb3ddI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$81\\cdot\\:\\frac{1}{81}\\ln\\left|x+3\\right|=\\ln\\left|x+3\\right|$$", "input": "81\\cdot\\:\\frac{1}{81}\\ln\\left|x+3\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:81}{81}\\ln\\left|x+3\\right|" }, { "type": "step", "primary": "Cancel the common factor: $$81$$", "result": "=\\ln\\left|x+3\\right|\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\ln\\left|x+3\\right|\\cdot\\:1=\\ln\\left|x+3\\right|$$", "result": "=\\ln\\left|x+3\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OxzBHNC/gxp2+CR/BZ0VwUJ4jayX35Tya8YzjotCwRSgTPYtU5MfChAqQ4fAEbNTVdNK6b/PmQukzNdLVQkh7Ba5wxbw0bDWKtupuEKqeB+sPlOTjVi6jQeHraQziu2f6ssEH0xyAo9npwPDb1r3Vh65A+CARNQWsE9xbiKjCZ8D2vk/DYOs8QR1k++gTUdgv1AviU7pPkkXpW7knX6Mmf7zwK10Exq2gj3OX9tYwzJ13L3hYrtLa5Vakyjnx6DU" } }, { "type": "interim", "title": "$$81\\cdot\\:\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|=\\ln\\left|4x^{2}-12x+36\\right|$$", "input": "81\\cdot\\:\\frac{1}{81}\\ln\\left|4x^{2}-12x+36\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:81}{81}\\ln\\left|4x^{2}-12x+36\\right|" }, { "type": "step", "primary": "Cancel the common factor: $$81$$", "result": "=\\ln\\left|4x^{2}-12x+36\\right|\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\ln\\left|4x^{2}-12x+36\\right|\\cdot\\:1=\\ln\\left|4x^{2}-12x+36\\right|$$", "result": "=\\ln\\left|4x^{2}-12x+36\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OxzBHNC/gxp2+CR/BZ0VwUJ4jayX35Tya8YzjotCwRTzqVGPWA3PbxOr6gWasVdeBuE28BmK7YuxDNoMg2D2JnWD310L1+P2yDQQfMEhENFuHwWD0XdpmBvfOlg6z1Fo9CJZVyEN6wQiS24Q6FQKcbEkq1kNtBdDDBl2tko0d0DZMTV/sawB5Tk8EZ+JVBvcWyzNTTBI7gBTqeCW2P0FNPX4Inu7SfRUTTlVzNfYg5vX8F1sxIwXhonVVlQtyqsEQQ6GT8MsfiVyKU2ABA9GBAzkXSQLf0jr9OV28gTIzZw=" } }, { "type": "step", "result": "=3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|+C", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "practiceLink": "/practice/integration-practice#area=main&subtopic=Substitution", "practiceTopic": "Integral Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=3\\ln\\left|x\\right|-\\ln\\left|x+3\\right|-\\ln\\left|4x^{2}-12x+36\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }