integral of (x^4+2x+6)/(x^3+x^2-2x)

{ "query": { "display": "$$\\int\\:\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{x^{2}}{2}-x-3\\ln\\left|x\\right|+3\\ln\\left|x-1\\right|+3\\ln\\left|x+2\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}dx=\\frac{x^{2}}{2}-x-3\\ln\\left|x\\right|+3\\ln\\left|x-1\\right|+3\\ln\\left|x+2\\right|+C$$", "input": "\\int\\:\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}dx", "steps": [ { "type": "interim", "title": "Take the partial fraction of $$\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}:{\\quad}x-1-\\frac{3}{x}+\\frac{3}{x-1}+\\frac{3}{x+2}$$", "input": "\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}", "steps": [ { "type": "interim", "title": "Long division $$\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}:{\\quad}x-1+\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}$$", "input": "\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}", "result": "=x-1+\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}", "steps": [ { "type": "interim", "title": "Divide $$\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}:{\\quad}\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}=x+\\frac{-x^{3}+2x^{2}+2x+6}{x^{3}+x^{2}-2x}$$", "result": "=x+\\frac{-x^{3}+2x^{2}+2x+6}{x^{3}+x^{2}-2x}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$x^{4}+2x+6$$<br/>and the divisor $$x^{3}+x^{2}-2x\\::\\:\\frac{x^{4}}{x^{3}}=x$$", "result": "\\mathrm{Quotient}=x" }, { "type": "step", "primary": "Multiply $$x^{3}+x^{2}-2x$$ by $$x:\\:x^{4}+x^{3}-2x^{2}$$", "secondary": [ "Subtract $$x^{4}+x^{3}-2x^{2}$$ from $$x^{4}+2x+6$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-x^{3}+2x^{2}+2x+6" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x^{4}+2x+6}{x^{3}+x^{2}-2x}=x+\\frac{-x^{3}+2x^{2}+2x+6}{x^{3}+x^{2}-2x}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-x^{3}+2x^{2}+2x+6}{x^{3}+x^{2}-2x}:{\\quad}\\frac{-x^{3}+2x^{2}+2x+6}{x^{3}+x^{2}-2x}=-1+\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}$$", "result": "=x-1+\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-x^{3}+2x^{2}+2x+6$$<br/>and the divisor $$x^{3}+x^{2}-2x\\::\\:\\frac{-x^{3}}{x^{3}}=-1$$", "result": "\\mathrm{Quotient}=-1" }, { "type": "step", "primary": "Multiply $$x^{3}+x^{2}-2x$$ by $$-1:\\:-x^{3}-x^{2}+2x$$", "secondary": [ "Subtract $$-x^{3}-x^{2}+2x$$ from $$-x^{3}+2x^{2}+2x+6$$ to get new remainder" ], "result": "\\mathrm{Remainder}=3x^{2}+6" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-x^{3}+2x^{2}+2x+6}{x^{3}+x^{2}-2x}=-1+\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } } ], "meta": { "solvingClass": "Long Division", "interimType": "Algebraic Manipulation Long Division Title 1Eq" } }, { "type": "step", "primary": "Continue partial fractions on remainder: $$\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}$$" }, { "type": "interim", "title": "Take the partial fraction of $$\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}:{\\quad}-\\frac{3}{x}+\\frac{3}{x-1}+\\frac{3}{x+2}$$", "input": "\\frac{3x^{2}+6}{x^{3}+x^{2}-2x}", "steps": [ { "type": "interim", "title": "Factor $$x^{3}+x^{2}-2x:{\\quad}x\\left(x-1\\right)\\left(x+2\\right)$$", "input": "x^{3}+x^{2}-2x", "result": "=\\frac{3x^{2}+6}{x\\left(x-1\\right)\\left(x+2\\right)}", "steps": [ { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x^{2}+x-2\\right)$$", "input": "x^{3}+x^{2}-2x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$", "$$x^{3}=x^{2}x$$" ], "result": "=x^{2}x+xx-2x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x^{2}+x-2\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=x\\left(x^{2}+x-2\\right)" }, { "type": "interim", "title": "Factor $$x^{2}+x-2:{\\quad}\\left(x-1\\right)\\left(x+2\\right)$$", "input": "x^{2}+x-2", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}+x-2", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=1,\\:c=-2$$", "$$u*v=-2,\\:u+v=1$$" ] }, { "type": "interim", "title": "Factors of $$2:{\\quad}1,\\:2$$", "input": "2", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpp3lFwBgr08J1hDIhHaqLjwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJmyKVlUUECPEUZsSDkCFf8u/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$2$$", "result": "1,\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$2:{\\quad}-1,\\:-2$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-2,\\:$$check if $$u+v=1$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-2:\\quad\\:u*v=-2,\\:u+v=-1\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=-1:\\quad\\:u*v=-2,\\:u+v=1\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=2,\\:v=-1" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-x\\right)+\\left(2x-2\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwn30q2MIJp/1sy1h0ZoW6o4HjZ0JmeAC3ZSEmMxWRNYu9GZPFaw2ewNA2aBnABa7Ku/snSOMzFBF/81Glop5NPlBgI/uz/WXQFrWuIqLHSyLITz6WHZx9UH0EGxKBGtUEBBOfyC7Me/gy/zVrVb5jy6lM1XdHV6wouRb0ZBuz4bhG4HL8yxsY3faiRjEFSmIgQ==" } }, { "type": "step", "result": "=\\left(x^{2}-x\\right)+\\left(2x-2\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-x:\\quad\\:x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$2\\:$$from $$2x-2:\\quad\\:2\\left(x-1\\right)$$", "input": "2x-2", "steps": [ { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7moABaoXIXq3E1MypH32OrsevWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G18aJzGOjaRVSiBHKVuIO6fKN6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17Xs0+6jPTos8h/1RIBPummW+tl6WFd7TeP+ieu4LzFGw2I=" } }, { "type": "step", "result": "=x\\left(x-1\\right)+2\\left(x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$x-1$$", "result": "=\\left(x-1\\right)\\left(x+2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=x\\left(x-1\\right)\\left(x+2\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Create the partial fraction template using the denominator $$x\\left(x-1\\right)\\left(x+2\\right)$$", "result": "\\frac{3x^{2}+6}{x\\left(x-1\\right)\\left(x+2\\right)}=\\frac{a_{0}}{x}+\\frac{a_{1}}{x-1}+\\frac{a_{2}}{x+2}", "steps": [ { "type": "step", "primary": "For $$x\\:$$add the partial fraction(s): $$\\frac{a_{0}}{x}$$" }, { "type": "step", "primary": "For $$x-1\\:$$add the partial fraction(s): $$\\frac{a_{1}}{x-1}$$" }, { "type": "step", "primary": "For $$x+2\\:$$add the partial fraction(s): $$\\frac{a_{2}}{x+2}$$" }, { "type": "step", "result": "\\frac{3x^{2}+6}{x\\left(x-1\\right)\\left(x+2\\right)}=\\frac{a_{0}}{x}+\\frac{a_{1}}{x-1}+\\frac{a_{2}}{x+2}" } ], "meta": { "interimType": "Partial Fraction Templates Top 1Eq" } }, { "type": "step", "primary": "Multiply equation by the denominator", "result": "\\frac{x\\left(3x^{2}+6\\right)\\left(x-1\\right)\\left(x+2\\right)}{x\\left(x-1\\right)\\left(x+2\\right)}=\\frac{a_{0}x\\left(x-1\\right)\\left(x+2\\right)}{x}+\\frac{a_{1}x\\left(x-1\\right)\\left(x+2\\right)}{x-1}+\\frac{a_{2}x\\left(x-1\\right)\\left(x+2\\right)}{x+2}" }, { "type": "step", "primary": "Simplify", "result": "3x^{2}+6=a_{0}\\left(x-1\\right)\\left(x+2\\right)+a_{1}x\\left(x+2\\right)+a_{2}x\\left(x-1\\right)" }, { "type": "step", "primary": "Solve the unknown parameters by plugging the real roots of the denominator: $$0,\\:1,\\:-2$$" }, { "type": "interim", "title": "For the denominator root $$0:{\\quad}a_{0}=-3$$", "steps": [ { "type": "step", "primary": "Plug in $$x=0\\:$$into the equation", "result": "3\\cdot\\:0^{2}+6=a_{0}\\left(0-1\\right)\\left(0+2\\right)+a_{1}\\cdot\\:0\\cdot\\:\\left(0+2\\right)+a_{2}\\cdot\\:0\\cdot\\:\\left(0-1\\right)" }, { "type": "step", "primary": "Expand", "result": "6=-2a_{0}" }, { "type": "interim", "title": "Solve $$6=-2a_{0}\\:$$for $$a_{0}:{\\quad}a_{0}=-3$$", "input": "6=-2a_{0}", "result": "a_{0}=-3", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-2a_{0}=6" }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2a_{0}=6", "result": "a_{0}=-3", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2a_{0}}{-2}=\\frac{6}{-2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-2a_{0}}{-2}=\\frac{6}{-2}", "result": "a_{0}=-3", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-2a_{0}}{-2}:{\\quad}a_{0}$$", "input": "\\frac{-2a_{0}}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2a_{0}}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=a_{0}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MVtvWib7/ZH4yzqtrtoPVGHRRwU7J6wSLWN8kxVMmG1wkKGJWEPFPk38sdJMsyPICVvPaHqz47+l4WBlTalmw03kCh3oevUunZ7/b0qFKBSBAc1PafP4ia+acEW7bvr5eyXKeiE3WgXEx9zGhk+6xg==" } }, { "type": "interim", "title": "Simplify $$\\frac{6}{-2}:{\\quad}-3$$", "input": "\\frac{6}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{6}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{2}=3$$", "result": "=-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7febBuae5wTwzbNy8yb78MC061ljBSPJeENOw2efoSWvW46RXHXnPJFddhpLU4qOFo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCB+9itPYKtzzCH6Tk1Z1q8tJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "a_{0}=-3" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "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 $$1:{\\quad}a_{1}=3$$", "steps": [ { "type": "step", "primary": "Plug in $$x=1\\:$$into the equation", "result": "3\\cdot\\:1^{2}+6=a_{0}\\left(1-1\\right)\\left(1+2\\right)+a_{1}\\cdot\\:1\\cdot\\:\\left(1+2\\right)+a_{2}\\cdot\\:1\\cdot\\:\\left(1-1\\right)" }, { "type": "step", "primary": "Expand", "result": "9=3a_{1}" }, { "type": "interim", "title": "Solve $$9=3a_{1}\\:$$for $$a_{1}:{\\quad}a_{1}=3$$", "input": "9=3a_{1}", "result": "a_{1}=3", "steps": [ { "type": "step", "primary": "Switch sides", "result": "3a_{1}=9" }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3a_{1}=9", "result": "a_{1}=3", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3a_{1}}{3}=\\frac{9}{3}" }, { "type": "step", "primary": "Simplify", "result": "a_{1}=3" } ], "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 $$-2:{\\quad}a_{2}=3$$", "steps": [ { "type": "step", "primary": "Plug in $$x=-2\\:$$into the equation", "result": "3\\left(-2\\right)^{2}+6=a_{0}\\left(\\left(-2\\right)-1\\right)\\left(\\left(-2\\right)+2\\right)+a_{1}\\left(-2\\right)\\left(\\left(-2\\right)+2\\right)+a_{2}\\left(-2\\right)\\left(\\left(-2\\right)-1\\right)" }, { "type": "step", "primary": "Expand", "result": "18=6a_{2}" }, { "type": "interim", "title": "Solve $$18=6a_{2}\\:$$for $$a_{2}:{\\quad}a_{2}=3$$", "input": "18=6a_{2}", "result": "a_{2}=3", "steps": [ { "type": "step", "primary": "Switch sides", "result": "6a_{2}=18" }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6a_{2}=18", "result": "a_{2}=3", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6a_{2}}{6}=\\frac{18}{6}" }, { "type": "step", "primary": "Simplify", "result": "a_{2}=3" } ], "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_{0}=-3,\\:a_{1}=3,\\:a_{2}=3" }, { "type": "step", "primary": "Plug the solutions to the partial fraction parameters to obtain the final result", "result": "\\frac{\\left(-3\\right)}{x}+\\frac{3}{x-1}+\\frac{3}{x+2}" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{3}{x}+\\frac{3}{x-1}+\\frac{3}{x+2}" } ], "meta": { "solvingClass": "Partial Fractions", "interimType": "Algebraic Manipulation Partial Fraction Top Title 1Eq" } }, { "type": "step", "result": "x-1-\\frac{3}{x}+\\frac{3}{x-1}+\\frac{3}{x+2}" } ], "meta": { "solvingClass": "Partial Fractions", "interimType": "Algebraic Manipulation Partial Fraction Top Title 1Eq" } }, { "type": "step", "result": "=\\int\\:x-1-\\frac{3}{x}+\\frac{3}{x-1}+\\frac{3}{x+2}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": "=\\int\\:xdx-\\int\\:1dx-\\int\\:\\frac{3}{x}dx+\\int\\:\\frac{3}{x-1}dx+\\int\\:\\frac{3}{x+2}dx" }, { "type": "interim", "title": "$$\\int\\:xdx=\\frac{x^{2}}{2}$$", "input": "\\int\\:xdx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=\\frac{x^{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{1+1}}{1+1}:{\\quad}\\frac{x^{2}}{2}$$", "input": "\\frac{x^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7814/6/Jz6acDoAMznrJ9GL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpPzIcDl+e6/8g9uDsiVdOq//YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:1dx=x$$", "input": "\\int\\:1dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{3}{x}dx=3\\ln\\left|x\\right|$$", "input": "\\int\\:\\frac{3}{x}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}dx" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{x}dx=\\ln\\left(\\left|x\\right|\\right)$$", "result": "=3\\ln\\left|x\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{3}{x-1}dx=3\\ln\\left|x-1\\right|$$", "input": "\\int\\:\\frac{3}{x-1}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-1}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{x-1}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: $$u=x-1$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x-1\\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(1\\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": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "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+4/5L7s7z4Oo03I1xk2hiqmTi2d1DMcjlLRK1jUV206qo4+vRN7yRnKSRJBHsnEXq1wS/zww7h8KUX9hN21wJE16H/NfZx2cv65xlj4FWO/jAv7Am1CptFvUOUfgDrM9m4ow9eu1Xql8XXPq6bNQlMm+36iNhkkjuzIgeJUg10ybKgq0r22txEId7lZcSHdTAsAvmTZFg==" } }, { "type": "step", "result": "=3\\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": "=3\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=x-1$$", "result": "=3\\ln\\left|x-1\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{3}{x+2}dx=3\\ln\\left|x+2\\right|$$", "input": "\\int\\:\\frac{3}{x+2}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}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{x+2}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: $$u=x+2$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x+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(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": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(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+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "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+4/5L7s73enLsVrmK9iTpShgEsS9mEcjlLRK1jUV206qo4+vRN7yRnKSRJBHsnEXq1wS/zww7h8KUX9hN21wJE16H/NfZx2cv65xlj4FWO/jAv7Am1CptFvUOUfgDrM9m4ow9eu1Xql8XXPq6bNQlMm+36iNhkkjuzIgeJUg10ybKgq0r22txEId7lZcSHdTAsAvmTZFg==" } }, { "type": "step", "result": "=3\\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": "=3\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=x+2$$", "result": "=3\\ln\\left|x+2\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{x^{2}}{2}-x-3\\ln\\left|x\\right|+3\\ln\\left|x-1\\right|+3\\ln\\left|x+2\\right|" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{x^{2}}{2}-x-3\\ln\\left|x\\right|+3\\ln\\left|x-1\\right|+3\\ln\\left|x+2\\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=\\frac{x^{2}}{2}-x-3\\ln\\left|x\\right|+3\\ln\\left|x-1\\right|+3\\ln\\left|x+2\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }