integral of (-8)/((x-2)(x^2+4))

{ "query": { "display": "$$\\int\\:\\frac{-8}{\\left(x-2\\right)\\left(x^{2}+4\\right)}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{-8}{(x-2)(x^{2}+4)}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan(\\frac{x}{2})+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{-8}{\\left(x-2\\right)\\left(x^{2}+4\\right)}dx=-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)+C$$", "input": "\\int\\:\\frac{-8}{\\left(x-2\\right)\\left(x^{2}+4\\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": "=-8\\cdot\\:\\int\\:\\frac{1}{\\left(x-2\\right)\\left(x^{2}+4\\right)}dx" }, { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{1}{\\left(x-2\\right)\\left(x^{2}+4\\right)}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: $$x=2u$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=4,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=2\\quad\\Rightarrow\\quad$$substitute $$x=2u$$" }, { "type": "interim", "title": "$$\\frac{dx}{du}=2$$", "input": "\\frac{d}{du}\\left(2u\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{du}{du}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{du}{du}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYvicOLv7Jdhyyrktzf6ZF9zZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51g8IPxS1V8HQ96mkP9ULxJR" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=2du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\left(2u-2\\right)\\left(\\left(2u\\right)^{2}+4\\right)}\\cdot\\:2du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\left(2u-2\\right)\\left(\\left(2u\\right)^{2}+4\\right)}\\cdot\\:2:{\\quad}\\frac{1}{4\\left(u-1\\right)\\left(u^{2}+1\\right)}$$", "input": "\\frac{1}{\\left(2u-2\\right)\\left(\\left(2u\\right)^{2}+4\\right)}\\cdot\\:2", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\left(2u-2\\right)\\left(\\left(2u\\right)^{2}+4\\right)}=\\frac{1}{\\left(2u-2\\right)\\left(4u^{2}+4\\right)}$$", "input": "\\frac{1}{\\left(2u-2\\right)\\left(\\left(2u\\right)^{2}+4\\right)}", "steps": [ { "type": "interim", "title": "$$\\left(2u\\right)^{2}=4u^{2}$$", "input": "\\left(2u\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cQDj2fG3PKYzjVjILAzgjs0ag8T1MwTer44+aCS/ZFAJ/+v6bFh9qlRZ7670glkkoX4usPSd1JXjYSuz11RWtEjZIe5ncy0wi4e8qtMXCVM=" } }, { "type": "step", "result": "=\\frac{1}{\\left(2u-2\\right)\\left(4u^{2}+4\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7a0OF+oKJCg5BpsKJmRaWooXFljXojxjGGXIkQjSFWSktOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmONu0gyM+xjt+6ZdFbZNH7uigCyMia7vQZHHIQ/pYIXuh//zb12i53htpqHknueEfaGseWT9zJN0NZ50jUMFlOZ8IMAXCLHYjlelR3L07P8UvDoa+MSjUH+n3w9lLmqCg+LCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{\\left(2u-2\\right)\\left(4u^{2}+4\\right)}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{\\left(2u-2\\right)\\left(4u^{2}+4\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{\\left(2u-2\\right)\\left(4u^{2}+4\\right)}" }, { "type": "interim", "title": "Factor $$\\left(2u-2\\right)\\left(4u^{2}+4\\right):{\\quad}8\\left(u-1\\right)\\left(u^{2}+1\\right)$$", "input": "\\left(2u-2\\right)\\left(4u^{2}+4\\right)", "result": "=\\frac{2}{8\\left(u-1\\right)\\left(u^{2}+1\\right)}", "steps": [ { "type": "interim", "title": "Factor $$2u-2:{\\quad}2\\left(u-1\\right)$$", "input": "2u-2", "result": "=2\\left(u-1\\right)\\left(4u^{2}+4\\right)", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2u-2\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(u-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$4u^{2}+4:{\\quad}4\\left(u^{2}+1\\right)$$", "input": "4u^{2}+4", "result": "=2\\left(u-1\\right)\\cdot\\:4\\left(u^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=4u^{2}+4\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(u^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Refine", "result": "=8\\left(u-1\\right)\\left(u^{2}+1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{4\\left(u-1\\right)\\left(u^{2}+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{4\\left(u-1\\right)\\left(u^{2}+1\\right)}du" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=-8\\cdot\\:\\int\\:\\frac{1}{4\\left(u-1\\right)\\left(u^{2}+1\\right)}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": "=-8\\cdot\\:\\frac{1}{4}\\cdot\\:\\int\\:\\frac{1}{\\left(u-1\\right)\\left(u^{2}+1\\right)}du" }, { "type": "interim", "title": "Take the partial fraction of $$\\frac{1}{\\left(u-1\\right)\\left(u^{2}+1\\right)}:{\\quad}\\frac{1}{2\\left(u-1\\right)}+\\frac{-u-1}{2\\left(u^{2}+1\\right)}$$", "input": "\\frac{1}{\\left(u-1\\right)\\left(u^{2}+1\\right)}", "steps": [ { "type": "interim", "title": "Create the partial fraction template using the denominator $$\\left(u-1\\right)\\left(u^{2}+1\\right)$$", "result": "\\frac{1}{\\left(u-1\\right)\\left(u^{2}+1\\right)}=\\frac{a_{0}}{u-1}+\\frac{a_{2}u+a_{1}}{u^{2}+1}", "steps": [ { "type": "step", "primary": "For $$u-1\\:$$add the partial fraction(s): $$\\frac{a_{0}}{u-1}$$" }, { "type": "step", "primary": "For $$u^{2}+1\\:$$add the partial fraction(s): $$\\frac{a_{2}u+a_{1}}{u^{2}+1}$$" }, { "type": "step", "result": "\\frac{1}{\\left(u-1\\right)\\left(u^{2}+1\\right)}=\\frac{a_{0}}{u-1}+\\frac{a_{2}u+a_{1}}{u^{2}+1}" } ], "meta": { "interimType": "Partial Fraction Templates Top 1Eq" } }, { "type": "step", "primary": "Multiply equation by the denominator", "result": "\\frac{1\\cdot\\:\\left(u-1\\right)\\left(u^{2}+1\\right)}{\\left(u-1\\right)\\left(u^{2}+1\\right)}=\\frac{a_{0}\\left(u-1\\right)\\left(u^{2}+1\\right)}{u-1}+\\frac{\\left(a_{2}u+a_{1}\\right)\\left(u-1\\right)\\left(u^{2}+1\\right)}{u^{2}+1}" }, { "type": "step", "primary": "Simplify", "result": "1=a_{0}\\left(u^{2}+1\\right)+\\left(a_{2}u+a_{1}\\right)\\left(u-1\\right)" }, { "type": "step", "primary": "Solve the unknown parameters by plugging the real roots of the denominator: $$1$$" }, { "type": "interim", "title": "For the denominator root $$1:{\\quad}a_{0}=\\frac{1}{2}$$", "steps": [ { "type": "step", "primary": "Plug in $$x=1\\:$$into the equation", "result": "1=a_{0}\\left(1^{2}+1\\right)+\\left(a_{2}\\cdot\\:1+a_{1}\\right)\\left(1-1\\right)" }, { "type": "step", "primary": "Expand", "result": "1=2a_{0}" }, { "type": "interim", "title": "Solve $$1=2a_{0}\\:$$for $$a_{0}:{\\quad}a_{0}=\\frac{1}{2}$$", "input": "1=2a_{0}", "result": "a_{0}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "2a_{0}=1" }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a_{0}=1", "result": "a_{0}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a_{0}}{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Simplify", "result": "a_{0}=\\frac{1}{2}" } ], "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}=\\frac{1}{2}" }, { "type": "step", "primary": "Plug in the solutions to the known parameters", "result": "1=\\frac{1}{2}\\left(u^{2}+1\\right)+\\left(a_{2}u+a_{1}\\right)\\left(u-1\\right)" }, { "type": "step", "primary": "Expand", "result": "1=\\frac{u^{2}}{2}+\\frac{1}{2}+a_{2}u^{2}-a_{2}u+a_{1}u-a_{1}" }, { "type": "step", "primary": "Extract Variables from within fractions", "result": "1=\\frac{1}{2}u^{2}+\\frac{1}{2}+a_{2}u^{2}-a_{2}u+a_{1}u-a_{1}" }, { "type": "step", "primary": "Group elements according to powers of $$x$$", "result": "1=u^{2}\\left(a_{2}+\\frac{1}{2}\\right)+u\\left(a_{1}-a_{2}\\right)+\\left(\\frac{1}{2}-a_{1}\\right)" }, { "type": "step", "primary": "Equate the coefficients of similar terms on both sides to create a list of equations", "result": "\\begin{bmatrix}\\frac{1}{2}-a_{1}=1\\\\-a_{2}+a_{1}=0\\\\\\frac{1}{2}+a_{2}=0\\end{bmatrix}" }, { "type": "interim", "title": "Solve system of equations:$${\\quad}a_{1}=-\\frac{1}{2},\\:a_{2}=-\\frac{1}{2}$$", "result": "a_{1}=-\\frac{1}{2},\\:a_{2}=-\\frac{1}{2}", "steps": [ { "type": "step", "result": "\\begin{bmatrix}\\frac{1}{2}-a_{1}=1\\\\-a_{2}+a_{1}=0\\\\\\frac{1}{2}+a_{2}=0\\end{bmatrix}" }, { "type": "interim", "title": "Isolate $$a_{1}\\:$$for $$\\frac{1}{2}-a_{1}=1:{\\quad}a_{1}=-\\frac{1}{2}$$", "input": "\\frac{1}{2}-a_{1}=1", "steps": [ { "type": "interim", "title": "Move $$\\frac{1}{2}\\:$$to the right side", "input": "\\frac{1}{2}-a_{1}=1", "result": "-a_{1}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{1}{2}$$ from both sides", "result": "\\frac{1}{2}-a_{1}-\\frac{1}{2}=1-\\frac{1}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1}{2}-a_{1}-\\frac{1}{2}=1-\\frac{1}{2}", "result": "-a_{1}=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{1}{2}-a_{1}-\\frac{1}{2}:{\\quad}-a_{1}$$", "input": "\\frac{1}{2}-a_{1}-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\frac{1}{2}-\\frac{1}{2}=0$$" }, { "type": "step", "result": "=-a_{1}" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$1-\\frac{1}{2}:{\\quad}\\frac{1}{2}$$", "input": "1-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{1\\cdot\\:2}{2}-\\frac{1}{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{1\\cdot\\:2-1}{2}" }, { "type": "interim", "title": "$$1\\cdot\\:2-1=1$$", "input": "1\\cdot\\:2-1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=2-1" }, { "type": "step", "primary": "Subtract the numbers: $$2-1=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZV4/pNEFJrYddVkM5e5CpN6GQqufR6tr2vPxOUv7H++P6Ubiv/bIrpol3G9QIK7hP/n/sT8Hudl/0KJRqY9qeRE27+eXjIuK5lxjSi/nI/o=" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DEWtR7xIevgCu8jrSCPX+QCWKUbvV6WK3fDUgFtg3Q/ysS6ztOAKKFAEXmiY4WQm1m0btACmSMMJpLyKhwdwNtbA+zX4bD3u3gx65o2NJhP7SQTBGPA8UkvSOcRK2w+PZ6NUR6U7LREw+iU0HKKk+w==" } }, { "type": "step", "result": "-a_{1}=\\frac{1}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYj91TdU3ptBGegltWpI9l1M9i9oAvf9GdVXNbR3Y4Fbq2xRNFceGPp0aewc9nbxefZAfd6GCL3c7/qgAIswOkjGAqomnF50Zs2t8jA+QD47lj3mae9LxmcMw5G2y+eGciXsI7YL9j/yxi5GNEDafsz7PbvGmKJwIuV6P6vcqqmDqs6cMsPSOyx1UIMZhE+2hVBG/30W+invWGzX2LbSU2qeVgQt3NwZg72jLCqp40vC7YIelMxfj+QZbkltLVbAv/llcpiRXYVHs43Doa/KCATffdwp61kZilyOEex8mi/vDzwJ/9PtsWe+NdQ1WJ9b580u6GVLaiCicFjOMFqAf2WVfsvBjVOwV8seK7dGuB8IRoujIgHyGv3ecmmpBRBbA7z5YFLBmMtI5s4SfwT5ssRDjI8CQv54DF5s1tSZ80gHItql3Plp18bHRXCdozdFwwVcBB87Yhd2KPHB2wvyOgocoxLjFgpPpvtAkQcbTFJkql0yvhvRnBlH+/q/XLk+JP0DIX52wVFV5c7HJNGxV7GFXw95YHalgjaNAfgc8DYnXYiBxEHSjvO+EJxohAWbkJMQVTIXxHbnuqrGIe5Uv+ELDIUjprCA+XIptxfb/EbOOB4CES9nGxD2S5ut+HDkiOxtWKyJwU+3om7eHGIbx8A80UCRHZyRKDzQfKBxnNPRTvJXrhRQTKZrV0QqJ1oeAGfve1FkLGYy7pfNHW7ge8PWtSON6otHROlKM8B29ZaSGtv0cR/lCZyOphCh0Gs1VlfpM+loOdtE+rK6eFfiDiSyldoqlBU6yoJSKeZv5uhdJ" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-a_{1}=\\frac{1}{2}", "result": "a_{1}=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-a_{1}}{-1}=\\frac{\\frac{1}{2}}{-1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-a_{1}}{-1}=\\frac{\\frac{1}{2}}{-1}", "result": "a_{1}=-\\frac{1}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-a_{1}}{-1}:{\\quad}a_{1}$$", "input": "\\frac{-a_{1}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{a_{1}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=a_{1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70IRh6NHfcxrkxqJYf9PMhTFSpzwRDPIFcbKdvhSPhgvMwViaLUXkeD+JukROhWdjJrv/g+oJRteZzxZArdSjdR429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJCF4OIwVF2UmzNW2gM3iBcvg==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{2}}{-1}:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{\\frac{1}{2}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{1}{2}}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "secondary": [ "$$\\frac{\\frac{1}{2}}{1}=\\frac{1}{2}$$" ], "result": "=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajTSN7ia5z2feuGtehi9xkJnNGoPE9TME3q+OPmgkv2RQrRKMQ0pA8A67jlmbMMkBFxUU/lo+nxKGmc+Xvlo3QeKY3ASC+aZqPN1DBWUUsybFLfFc+K4uSP6MmS0QGPb/Uu55bg7XYVkkJ9Qa7Xc3lSC/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "a_{1}=-\\frac{1}{2}" } ], "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 $$a_{1}=-\\frac{1}{2}$$", "result": "\\begin{bmatrix}\\frac{1}{2}+a_{2}=0\\\\-a_{2}-\\frac{1}{2}=0\\end{bmatrix}" }, { "type": "interim", "title": "Isolate $$a_{2}\\:$$for $$\\frac{1}{2}+a_{2}=0:{\\quad}a_{2}=-\\frac{1}{2}$$", "input": "\\frac{1}{2}+a_{2}=0", "steps": [ { "type": "interim", "title": "Move $$\\frac{1}{2}\\:$$to the right side", "input": "\\frac{1}{2}+a_{2}=0", "result": "a_{2}=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{1}{2}$$ from both sides", "result": "\\frac{1}{2}+a_{2}-\\frac{1}{2}=0-\\frac{1}{2}" }, { "type": "step", "primary": "Simplify", "result": "a_{2}=-\\frac{1}{2}" } ], "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 $$a_{2}=-\\frac{1}{2}$$", "result": "\\begin{bmatrix}-\\left(-\\frac{1}{2}\\right)-\\frac{1}{2}=0\\end{bmatrix}" }, { "type": "interim", "title": "Simplify", "input": "-\\left(-\\frac{1}{2}\\right)-\\frac{1}{2}=0", "steps": [ { "type": "interim", "title": "$$-\\left(-\\frac{1}{2}\\right)-\\frac{1}{2}=0$$", "input": "-\\left(-\\frac{1}{2}\\right)-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1}{2}-\\frac{1}{2}" }, { "type": "step", "primary": "Add similar elements: $$\\frac{1}{2}-\\frac{1}{2}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e1C1weT57OzLTbntetXUIF4oPWJY2gxIeILHNkJNmCwtOtZYwUjyXhDTsNnn6ElrUWT0a35DQ5ovrM5AARYqVDG9UEt9Wgb2B8Ci141JAG+ZgvabLpGANmKkqxJe4aCmJLtQwh+ptZafNrOeRiJ45g==" } }, { "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_{1}=-\\frac{1}{2},\\:a_{2}=-\\frac{1}{2}" } ], "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}{2}}{u-1}+\\frac{\\left(-\\frac{1}{2}\\right)u+\\left(-\\frac{1}{2}\\right)}{u^{2}+1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\frac{1}{2}}{u-1}+\\frac{\\left(-\\frac{1}{2}\\right)u+\\left(-\\frac{1}{2}\\right)}{u^{2}+1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{2}}{u-1}:{\\quad}\\frac{1}{2\\left(u-1\\right)}$$", "input": "\\frac{\\frac{1}{2}}{u-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{2\\left(u-1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajW1u4/Vh6Tx6BDwJjmqyGAPdd47a0hQ8flDbGsI5To1doDDfDl9rb93jWiqTfsVP2mhmEj7CpnVQ//E2lwBxWhnWwPs1+Gw97t4MeuaNjSYTwPBBw3byB5RPtpSy6YLt2kMDxPXzX1uYobf/HEieXaOCJ0W6EudIoEVBtOTE+NYm" } }, { "type": "interim", "title": "Simplify $$\\frac{\\left(-\\frac{1}{2}\\right)u+\\left(-\\frac{1}{2}\\right)}{u^{2}+1}:{\\quad}\\frac{-u-1}{2\\left(u^{2}+1\\right)}$$", "input": "\\frac{\\left(-\\frac{1}{2}\\right)u+\\left(-\\frac{1}{2}\\right)}{u^{2}+1}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-\\frac{1}{2}u-\\frac{1}{2}}{u^{2}+1}" }, { "type": "interim", "title": "$$\\frac{1}{2}u=\\frac{u}{2}$$", "input": "\\frac{1}{2}u", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:u}{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:u=u$$", "result": "=\\frac{u}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s761hg8fRasG0fSZdeTd5OAy061ljBSPJeENOw2efoSWtIanH17cuiWHdaViIEN2Y4OxA7Ju3BsHRdTMg0E9WsxfpBZKlms0Ic3JydKPoSmTAPbD9qrmQR8eGWP5QDYm0FjwE87HTCWyAU3ypRroDMDQ==" } }, { "type": "step", "result": "=\\frac{-\\frac{u}{2}-\\frac{1}{2}}{u^{2}+1}" }, { "type": "interim", "title": "Combine the fractions $$-\\frac{u}{2}-\\frac{1}{2}:{\\quad}\\frac{-u-1}{2}$$", "result": "=\\frac{\\frac{-u-1}{2}}{u^{2}+1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-u-1}{2}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{-u-1}{2\\left(u^{2}+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fKlREPZ8+xcP7ahexfiv7KHBXiUXfyqKTzIQzwEVgVXM80DLbZ/tYjfm37Qhzncj/KtF2sjFUSIoehkmXzUmKKORWLXkjysF58uLgjK3bCsqClFHXxVvadf6SU+UEvxnzs8cOrAGllCgJIUzAiTta3ql8XXPq6bNQlMm+36iNhljgtURsNZ8mF2q2lQDr86HeehhjBIcgzCrORRyeOWM7uNDkraK9Q52wYVgseUm62vGpa2JLN1TmWtD1MsxrX3W" } }, { "type": "step", "result": "=\\frac{1}{2\\left(u-1\\right)}+\\frac{-u-1}{2\\left(u^{2}+1\\right)}" } ], "meta": { "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\frac{1}{2\\left(u-1\\right)}+\\frac{-u-1}{2\\left(u^{2}+1\\right)}" } ], "meta": { "solvingClass": "Partial Fractions", "interimType": "Algebraic Manipulation Partial Fraction Top Title 1Eq" } }, { "type": "step", "result": "=-8\\cdot\\:\\frac{1}{4}\\cdot\\:\\int\\:\\frac{1}{2\\left(u-1\\right)}+\\frac{-u-1}{2\\left(u^{2}+1\\right)}du" }, { "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": "=-8\\cdot\\:\\frac{1}{4}\\left(\\int\\:\\frac{1}{2\\left(u-1\\right)}du+\\int\\:\\frac{-u-1}{2\\left(u^{2}+1\\right)}du\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{2\\left(u-1\\right)}du=\\frac{1}{2}\\ln\\left|u-1\\right|$$", "input": "\\int\\:\\frac{1}{2\\left(u-1\\right)}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": "=\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{u-1}du" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{u-1}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: $$v=u-1$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=1$$", "input": "\\frac{d}{du}\\left(u-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{du}{du}-\\frac{d}{du}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{du}{du}=1$$", "input": "\\frac{du}{du}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{du}{du}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYipXfuvZQYcWZ3RXZmhAT9ljqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIfqIOlxNXEONDm3M0PlIv9pOXvV+QvzGT1U5/bJzrRe1" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(1\\right)=0$$", "input": "\\frac{d}{du}\\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+idP5vLVrjUll6eMdYoonkTa3K+3a3OJF7IGZShVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTu3gZlAnOLgeYDts0TT4yEK" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=1du$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{v}\\cdot\\:1dv" }, { "type": "step", "result": "=\\int\\:\\frac{1}{v}dv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70gI8jAqlW/NH2yNxUDsGVccjlLRK1jUV206qo4+vRN7yRnKSRJBHsnEXq1wS/zww7h8KUX9hN21wJE16H/NfZx2cv65xlj4FWO/jAv7Am1C+/JFDwJFx/NFDc7dwwjRB3ql8XXPq6bNQlMm+36iNhkkjuzIgeJUg10ybKgq0r22txEId7lZcSHdTAsAvmTZFg==" } }, { "type": "step", "result": "=\\frac{1}{2}\\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": "=\\frac{1}{2}\\ln\\left|v\\right|" }, { "type": "step", "primary": "Substitute back $$v=u-1$$", "result": "=\\frac{1}{2}\\ln\\left|u-1\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{-u-1}{2\\left(u^{2}+1\\right)}du=\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|u^{2}+1\\right|-\\arctan\\left(u\\right)\\right)$$", "input": "\\int\\:\\frac{-u-1}{2\\left(u^{2}+1\\right)}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": "=\\frac{1}{2}\\cdot\\:\\int\\:\\frac{-u-1}{u^{2}+1}du" }, { "type": "interim", "title": "Expand $$\\frac{-u-1}{u^{2}+1}:{\\quad}-\\frac{u}{u^{2}+1}-\\frac{1}{u^{2}+1}$$", "input": "\\frac{-u-1}{u^{2}+1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "result": "=-\\frac{u}{u^{2}+1}-\\frac{1}{u^{2}+1}" } ], "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}{2}\\left(-\\int\\:\\frac{u}{u^{2}+1}du-\\int\\:\\frac{1}{u^{2}+1}du\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{u}{u^{2}+1}du=\\frac{1}{2}\\ln\\left|u^{2}+1\\right|$$", "input": "\\int\\:\\frac{u}{u^{2}+1}du", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{u}{u^{2}+1}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: $$v=u^{2}+1$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=2u$$", "input": "\\frac{d}{du}\\left(u^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{du}\\left(u^{2}\\right)+\\frac{d}{du}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(1\\right)=0$$", "input": "\\frac{d}{du}\\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+idP5vLVrjUll6eMdYoonkTa3K+3a3OJF7IGZShVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTu3gZlAnOLgeYDts0TT4yEK" } }, { "type": "step", "result": "=2u+0" }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=2udu$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{1}{2u}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{u}{v}\\cdot\\:\\frac{1}{2u}dv" }, { "type": "interim", "title": "Simplify $$\\frac{u}{v}\\cdot\\:\\frac{1}{2u}:{\\quad}\\frac{1}{2v}$$", "input": "\\frac{u}{v}\\cdot\\:\\frac{1}{2u}", "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\\:2u}" }, { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=\\frac{1}{v\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{2v}dv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75epHp60CIcwFmRo4avNVsw9kUUDxBYkccTMjOJZAqit2NCwimM7dB8C524HvMCB4ymZOc9q9xxqJAg2jt99wha4eveJYYFfQMHyfOzMWdYhzzpAOUccJvS8EUrp1nHLzEUqTd96MWTKI6Kr2Ib0iQBZegS2gwh8pq/gwNfwaDRbAwNT33I9ftOGdlyhcHnXlA==" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{2v}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": "=\\frac{1}{2}\\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": "=\\frac{1}{2}\\ln\\left|v\\right|" }, { "type": "step", "primary": "Substitute back $$v=u^{2}+1$$", "result": "=\\frac{1}{2}\\ln\\left|u^{2}+1\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{u^{2}+1}du=\\arctan\\left(u\\right)$$", "input": "\\int\\:\\frac{1}{u^{2}+1}du", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u^{2}+1}du=\\arctan\\left(u\\right)$$", "result": "=\\arctan\\left(u\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74gU8hoOQW19c4L6URPT0ws9kUUDxBYkccTMjOJZAqitflaC/Xf7yHEDLIJDAX9Vh9XCTC4lVL+JHElO8ae5xgsMah7G9TYccA7L78P0RKhe45jiv5feQ19Y6WEwl98RgoDH91adNlWJgHE7vImUIYAl6z7MUvGxlhUBodbyY9sZ" } }, { "type": "step", "result": "=\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|u^{2}+1\\right|-\\arctan\\left(u\\right)\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-8\\cdot\\:\\frac{1}{4}\\left(\\frac{1}{2}\\ln\\left|u-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|u^{2}+1\\right|-\\arctan\\left(u\\right)\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\frac{x}{2}$$", "result": "=-8\\cdot\\:\\frac{1}{4}\\left(\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{x}{2}\\right)^{2}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-8\\cdot\\:\\frac{1}{4}\\left(\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{x}{2}\\right)^{2}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\right):{\\quad}-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)$$", "input": "-8\\cdot\\:\\frac{1}{4}\\left(\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{x}{2}\\right)^{2}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\right)", "result": "=-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:8}{4}\\left(\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{x}{2}\\right)^{2}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:8}{4}=2$$", "input": "\\frac{1\\cdot\\:8}{4}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:8=8$$", "result": "=\\frac{8}{4}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{4}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCtLr/RFPvs/lk4+/d72l0PGrju+5Z51e/ZZSD3gRHwjBSLdINoPD2MyLmUXT+YnlMgZ7aVYt9I8x8R7JWUC2axBer4ub/f1i2PfGoPbJtOhu" } }, { "type": "step", "result": "=-2\\left(\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{x}{2}\\right)^{2}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\left(\\frac{x}{2}\\right)^{2}=\\frac{x^{2}}{4}$$", "input": "\\left(\\frac{x}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{x^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{x^{2}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qn3wXiF1uFa2Fg/ESc6hHY5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjb6dttZphwDk/C+i0kC3tgv8//6/nV5O4fb8Xgwi7maodBxm/MDYCEt+AuSiG1vb6bPisUk8ilRa4au3dTpXClLsav6oKwOalk3ENSmN9wNU=" } }, { "type": "step", "result": "=-2\\left(\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\right)" }, { "type": "step", "result": "=-2\\left(\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-2,\\:b=\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|,\\:c=\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)$$" ], "result": "=-2\\cdot\\:\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|+\\left(-2\\right)\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-2\\cdot\\:\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|-2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-2\\cdot\\:\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|-2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right):{\\quad}-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)$$", "input": "-2\\cdot\\:\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|-2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)", "result": "=-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|=\\ln\\left|\\frac{x}{2}-1\\right|$$", "input": "2\\cdot\\:\\frac{1}{2}\\ln\\left|\\frac{x}{2}-1\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}\\ln\\left|\\frac{x}{2}-1\\right|" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\ln\\left|\\frac{x}{2}-1\\right|\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\ln\\left|\\frac{x}{2}-1\\right|\\cdot\\:1=\\ln\\left|\\frac{x}{2}-1\\right|$$", "result": "=\\ln\\left|\\frac{x}{2}-1\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvikpoPwoVN8CD9sMh9OrOY+AbR4pgEQnn+qhPakHuErx07rqvwAa1tYGHAzXjzk+GoQlAlm5MBjmOz6iqN1PySyV44dg+mQ2PAMAPvXs6gI/NRMm7+y+ygNWrtq7LzjhtObEkq1kNtBdDDBl2tko0d0BSqCkHW72ErKWThqEDXOHOvwsWM+AXenOAHy2jjpmkzZbiFFIUen+mX8ZzWSbseH2S+l8paeg0NTh/dEn/CuzToZOwukHDds8bfxWU/wRJJcfLMHliO7wZJNuHIIoRAWk=" } }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)=-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)$$", "input": "2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)\\cdot\\:1" }, { "type": "step", "primary": "Refine", "result": "=-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=-\\ln\\left|\\frac{x}{2}-1\\right|-\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)" }, { "type": "interim", "title": "$$-\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right):{\\quad}\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)$$", "input": "-\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|-\\arctan\\left(\\frac{x}{2}\\right)\\right)", "result": "=-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(-\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|\\right)-\\left(-\\arctan\\left(\\frac{x}{2}\\right)\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{x}{2}\\right)" } ], "meta": { "interimType": "N/A" } } ], "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": "=-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan\\left(\\frac{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=-\\ln\\left|\\frac{x}{2}-1\\right|+\\frac{1}{2}\\ln\\left|\\frac{x^{2}}{4}+1\\right|+\\arctan(\\frac{x}{2})+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }