integral of (4x)/((x-1)^2(x+1))

{ "query": { "display": "$$\\int\\:\\frac{4x}{\\left(x-1\\right)^{2}\\left(x+1\\right)}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{4x}{(x-1)^{2}(x+1)}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{4x}{\\left(x-1\\right)^{2}\\left(x+1\\right)}dx=\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|+C$$", "input": "\\int\\:\\frac{4x}{\\left(x-1\\right)^{2}\\left(x+1\\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": "=4\\cdot\\:\\int\\:\\frac{x}{\\left(x-1\\right)^{2}\\left(x+1\\right)}dx" }, { "type": "interim", "title": "Take the partial fraction of $$\\frac{x}{\\left(x-1\\right)^{2}\\left(x+1\\right)}:{\\quad}\\frac{1}{4\\left(x-1\\right)}+\\frac{1}{2\\left(x-1\\right)^{2}}-\\frac{1}{4\\left(x+1\\right)}$$", "input": "\\frac{x}{\\left(x-1\\right)^{2}\\left(x+1\\right)}", "steps": [ { "type": "interim", "title": "Create the partial fraction template using the denominator $$\\left(x-1\\right)^{2}\\left(x+1\\right)$$", "result": "\\frac{x}{\\left(x-1\\right)^{2}\\left(x+1\\right)}=\\frac{a_{0}}{x-1}+\\frac{a_{1}}{\\left(x-1\\right)^{2}}+\\frac{a_{2}}{x+1}", "steps": [ { "type": "step", "primary": "For $$\\left(x-1\\right)^{2}\\:$$add the partial fraction(s): $$\\frac{a_{0}}{x-1}+\\frac{a_{1}}{\\left(x-1\\right)^{2}}$$" }, { "type": "step", "primary": "For $$x+1\\:$$add the partial fraction(s): $$\\frac{a_{2}}{x+1}$$" }, { "type": "step", "result": "\\frac{x}{\\left(x-1\\right)^{2}\\left(x+1\\right)}=\\frac{a_{0}}{x-1}+\\frac{a_{1}}{\\left(x-1\\right)^{2}}+\\frac{a_{2}}{x+1}" } ], "meta": { "interimType": "Partial Fraction Templates Top 1Eq" } }, { "type": "step", "primary": "Multiply equation by the denominator", "result": "\\frac{x\\left(x-1\\right)^{2}\\left(x+1\\right)}{\\left(x-1\\right)^{2}\\left(x+1\\right)}=\\frac{a_{0}\\left(x-1\\right)^{2}\\left(x+1\\right)}{x-1}+\\frac{a_{1}\\left(x-1\\right)^{2}\\left(x+1\\right)}{\\left(x-1\\right)^{2}}+\\frac{a_{2}\\left(x-1\\right)^{2}\\left(x+1\\right)}{x+1}" }, { "type": "step", "primary": "Simplify", "result": "x=a_{0}\\left(x-1\\right)\\left(x+1\\right)+a_{1}\\left(x+1\\right)+a_{2}\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Solve the unknown parameters by plugging the real roots of the denominator: $$1,\\:-1$$" }, { "type": "interim", "title": "For the denominator root $$1:{\\quad}a_{1}=\\frac{1}{2}$$", "steps": [ { "type": "step", "primary": "Plug in $$x=1\\:$$into the equation", "result": "1=a_{0}\\left(1-1\\right)\\left(1+1\\right)+a_{1}\\left(1+1\\right)+a_{2}\\left(1-1\\right)^{2}" }, { "type": "step", "primary": "Expand", "result": "1=2a_{1}" }, { "type": "interim", "title": "Solve $$1=2a_{1}\\:$$for $$a_{1}:{\\quad}a_{1}=\\frac{1}{2}$$", "input": "1=2a_{1}", "result": "a_{1}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "2a_{1}=1" }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a_{1}=1", "result": "a_{1}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a_{1}}{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Simplify", "result": "a_{1}=\\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": "interim", "title": "For the denominator root $$-1:{\\quad}a_{2}=-\\frac{1}{4}$$", "steps": [ { "type": "step", "primary": "Plug in $$x=-1\\:$$into the equation", "result": "\\left(-1\\right)=a_{0}\\left(\\left(-1\\right)-1\\right)\\left(\\left(-1\\right)+1\\right)+a_{1}\\left(\\left(-1\\right)+1\\right)+a_{2}\\left(\\left(-1\\right)-1\\right)^{2}" }, { "type": "step", "primary": "Expand", "result": "-1=4a_{2}" }, { "type": "interim", "title": "Solve $$-1=4a_{2}\\:$$for $$a_{2}:{\\quad}a_{2}=-\\frac{1}{4}$$", "input": "-1=4a_{2}", "result": "a_{2}=-\\frac{1}{4}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "4a_{2}=-1" }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4a_{2}=-1", "result": "a_{2}=-\\frac{1}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4a_{2}}{4}=\\frac{-1}{4}" }, { "type": "step", "primary": "Simplify", "result": "a_{2}=-\\frac{1}{4}" } ], "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_{1}=\\frac{1}{2},\\:a_{2}=-\\frac{1}{4}" }, { "type": "step", "primary": "Plug in the solutions to the known parameters", "result": "x=a_{0}\\left(x-1\\right)\\left(x+1\\right)+\\frac{1}{2}\\left(x+1\\right)+\\left(-\\frac{1}{4}\\right)\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Expand", "result": "x=-a_{0}+a_{0}x^{2}-\\frac{x^{2}}{4}+x+\\frac{1}{4}" }, { "type": "step", "primary": "Extract Variables from within fractions", "result": "x=-a_{0}+a_{0}x^{2}-\\frac{1}{4}x^{2}+x+\\frac{1}{4}" }, { "type": "step", "primary": "Group elements according to powers of $$x$$", "result": "1\\cdot\\:x=x^{2}\\left(a_{0}-\\frac{1}{4}\\right)+1\\cdot\\:x+\\left(\\frac{1}{4}-a_{0}\\right)" }, { "type": "step", "primary": "Solve $$-a_{0}+\\frac{1}{4}=0\\:$$for $$a_{0}$$", "result": "a_{0}=\\frac{1}{4}" }, { "type": "step", "primary": "Plug the solutions to the partial fraction parameters to obtain the final result", "result": "\\frac{\\frac{1}{4}}{x-1}+\\frac{\\frac{1}{2}}{\\left(x-1\\right)^{2}}+\\frac{\\left(-\\frac{1}{4}\\right)}{x+1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\frac{1}{4}}{x-1}+\\frac{\\frac{1}{2}}{\\left(x-1\\right)^{2}}+\\frac{\\left(-\\frac{1}{4}\\right)}{x+1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{4}}{x-1}:{\\quad}\\frac{1}{4\\left(x-1\\right)}$$", "input": "\\frac{\\frac{1}{4}}{x-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{4\\left(x-1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajduE9ISdtEwZhPJkr0UaHd7dd47a0hQ8flDbGsI5To1doDDfDl9rb93jWiqTfsVP2jF9yTu6Q83DoiXwk077jHzWwPs1+Gw97t4MeuaNjSYTwPBBw3byB5RPtpSy6YLt2pVPkaoCirv+JbXf4bHBM/ZjGG9JkyBKayXF2wfKXSEb" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{2}}{\\left(x-1\\right)^{2}}:{\\quad}\\frac{1}{2\\left(x-1\\right)^{2}}$$", "input": "\\frac{\\frac{1}{2}}{\\left(x-1\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{2\\left(x-1\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajc6oVoBAJ9GvRwl00GEIctp8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcaujpVBE40deap5Y/ipgTM7kRqzjWG0dFM/TWnjzf0PnlQ7kAjP76qW66lOUsURwT0nek7dhlaN3dpRMO5SBsK+5TGZn8oR9o3T1e7vRqfg4W8pXXVBzYmEcbFxkcAsY6b8g==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\left(-\\frac{1}{4}\\right)}{x+1}:{\\quad}-\\frac{1}{4\\left(x+1\\right)}$$", "input": "\\frac{-\\frac{1}{4}}{x+1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{1}{4}}{x+1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{1}{4}}{x+1}=\\frac{1}{4\\left(x+1\\right)}$$" ], "result": "=-\\frac{1}{4\\left(x+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVew80q2mTjsbrYL56bIYAgYgJ/ZZA32ZInFBpDtxBfiKmWiTEpQjat3SO7/l2m58l8MHy/WYPWVmRrBPQqZY03GBBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2dRGAYbzTdnt9bjeC+PR3qBbxrWJ+7kU+Qp+0ey1/atg" } }, { "type": "step", "result": "=\\frac{1}{4\\left(x-1\\right)}+\\frac{1}{2\\left(x-1\\right)^{2}}-\\frac{1}{4\\left(x+1\\right)}" } ], "meta": { "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\frac{1}{4\\left(x-1\\right)}+\\frac{1}{2\\left(x-1\\right)^{2}}-\\frac{1}{4\\left(x+1\\right)}" } ], "meta": { "solvingClass": "Partial Fractions", "interimType": "Algebraic Manipulation Partial Fraction Top Title 1Eq" } }, { "type": "step", "result": "=4\\cdot\\:\\int\\:\\frac{1}{4\\left(x-1\\right)}+\\frac{1}{2\\left(x-1\\right)^{2}}-\\frac{1}{4\\left(x+1\\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": "=4\\left(\\int\\:\\frac{1}{4\\left(x-1\\right)}dx+\\int\\:\\frac{1}{2\\left(x-1\\right)^{2}}dx-\\int\\:\\frac{1}{4\\left(x+1\\right)}dx\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{4\\left(x-1\\right)}dx=\\frac{1}{4}\\ln\\left|x-1\\right|$$", "input": "\\int\\:\\frac{1}{4\\left(x-1\\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}{4}\\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": "=\\frac{1}{4}\\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}{4}\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=x-1$$", "result": "=\\frac{1}{4}\\ln\\left|x-1\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{2\\left(x-1\\right)^{2}}dx=-\\frac{1}{2\\left(x-1\\right)}$$", "input": "\\int\\:\\frac{1}{2\\left(x-1\\right)^{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": "=\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{\\left(x-1\\right)^{2}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{\\left(x-1\\right)^{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: $$u=x-1$$" ] }, { "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^{2}}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{2}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71ee3Sz0BjE8PvOH8Rt1UQxDTyETaNhQhbj2klFLXAi+cSPKmf0CNKWC8kGIkCk3X3KtxwM1n7owdoG2GFz6ksdsfL2GYVRvzzfVlswjxO2YYk+kwx0fumEC/w52tX+Pm/UZDHRRVkrDOQpTZTNW6opSBv6izheLVUKQ/emokAUyEKC4SmDfgdaB0h/PWlULm7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{u^{2}}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{u^{2}}du", "result": "=\\frac{1}{2}\\left(-\\frac{1}{u}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{u^{2}}=u^{-2}$$" ], "result": "=\\int\\:u^{-2}du", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{-2+1}}{-2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{-2+1}}{-2+1}:{\\quad}-\\frac{1}{u}$$", "input": "\\frac{u^{-2+1}}{-2+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=\\frac{u^{-1}}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{u^{-1}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-u^{-1}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-1}=\\frac{1}{a}$$", "result": "=-\\frac{1}{u}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{u}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/RSr02Agv0MR/qV7Nm+eMMy4+rY5ULRUEksemusM4Yyrrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx4OQzbEnsuafNY3nX9QxDlJ1HXTSqqQEjS1gpf6I+JyHQS4M5VpC8qh+oehjmM1qmweKkh+28FiXwy+Vsz8xLQiialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "Substitute back $$u=x-1$$", "result": "=\\frac{1}{2}\\left(-\\frac{1}{x-1}\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\left(-\\frac{1}{x-1}\\right):{\\quad}-\\frac{1}{2\\left(x-1\\right)}$$", "input": "\\frac{1}{2}\\left(-\\frac{1}{x-1}\\right)", "result": "=-\\frac{1}{2\\left(x-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2}\\cdot\\:\\frac{1}{x-1}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{1\\cdot\\:1}{2\\left(x-1\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=-\\frac{1}{2\\left(x-1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79jHpDL2BX3rbBpw0SEa3inpqDjtQ3rrC8j3OIElIWjwtOtZYwUjyXhDTsNnn6ElryOsg4xTbsj8PJfnagYu7Q56RiyfbDFAmwNE0lhhvvD+jeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIF0KZmJHf2qZbg5+r2CmRPJjvcHafgAxq1Y3psT0lJMsJLd1ohke2Wgml78++2zI0g==" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{4\\left(x+1\\right)}dx=\\frac{1}{4}\\ln\\left|x+1\\right|$$", "input": "\\int\\:\\frac{1}{4\\left(x+1\\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}{4}\\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+4/5L7s7xUNxvMK5+mw478WPt4Ez1YcjlLRK1jUV206qo4+vRN7yRnKSRJBHsnEXq1wS/zww7h8KUX9hN21wJE16H/NfZx2cv65xlj4FWO/jAv7Am1CptFvUOUfgDrM9m4ow9eu1Xql8XXPq6bNQlMm+36iNhkkjuzIgeJUg10ybKgq0r22txEId7lZcSHdTAsAvmTZFg==" } }, { "type": "step", "result": "=\\frac{1}{4}\\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}{4}\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=x+1$$", "result": "=\\frac{1}{4}\\ln\\left|x+1\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=4\\left(\\frac{1}{4}\\ln\\left|x-1\\right|-\\frac{1}{2\\left(x-1\\right)}-\\frac{1}{4}\\ln\\left|x+1\\right|\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(\\frac{1}{4}\\ln\\left|x-1\\right|-\\frac{1}{2\\left(x-1\\right)}-\\frac{1}{4}\\ln\\left|x+1\\right|\\right):{\\quad}\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|$$", "input": "4\\left(\\frac{1}{4}\\ln\\left|x-1\\right|-\\frac{1}{2\\left(x-1\\right)}-\\frac{1}{4}\\ln\\left|x+1\\right|\\right)", "result": "=\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=4\\cdot\\:\\frac{1}{4}\\ln\\left|x-1\\right|+4\\left(-\\frac{1}{2\\left(x-1\\right)}\\right)+4\\left(-\\frac{1}{4}\\ln\\left|x+1\\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": "=4\\cdot\\:\\frac{1}{4}\\ln\\left|x-1\\right|-4\\cdot\\:\\frac{1}{2\\left(x-1\\right)}-4\\cdot\\:\\frac{1}{4}\\ln\\left|x+1\\right|" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:\\frac{1}{4}\\ln\\left|x-1\\right|-4\\cdot\\:\\frac{1}{2\\left(x-1\\right)}-4\\cdot\\:\\frac{1}{4}\\ln\\left|x+1\\right|:{\\quad}\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|$$", "input": "4\\cdot\\:\\frac{1}{4}\\ln\\left|x-1\\right|-4\\cdot\\:\\frac{1}{2\\left(x-1\\right)}-4\\cdot\\:\\frac{1}{4}\\ln\\left|x+1\\right|", "result": "=\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|", "steps": [ { "type": "interim", "title": "$$4\\cdot\\:\\frac{1}{4}\\ln\\left|x-1\\right|=\\ln\\left|x-1\\right|$$", "input": "4\\cdot\\:\\frac{1}{4}\\ln\\left|x-1\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{4}\\ln\\left|x-1\\right|" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\ln\\left|x-1\\right|\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\ln\\left|x-1\\right|\\cdot\\:1=\\ln\\left|x-1\\right|$$", "result": "=\\ln\\left|x-1\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV8/8hEWEUvNRhNZaQUynkRETppdm7s+eWCmxZgjTj4J4/ICf2WQN9mSJxQaQ7cQX4it8wm4YsT3c+x2f2pAHISNxjZdLJpHjx3iQqK5De8h48HRHdzS5g2HJFanH3nVRLiHiXj0Ip1doXRPIJCQFOu2TNOFCXICZb1KuA1iHGPUykjoZYr/6mXnYDx8qKaA8+zNjiYxHL9VaXvRlpoAK1RRywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$4\\cdot\\:\\frac{1}{2\\left(x-1\\right)}=\\frac{2}{x-1}$$", "input": "4\\cdot\\:\\frac{1}{2\\left(x-1\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{2\\left(x-1\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=\\frac{4}{2\\left(x-1\\right)}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=\\frac{2}{\\left(x-1\\right)}" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\frac{2}{x-1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV8/HOlcu0JDPiGgJy0Qj916/dd47a0hQ8flDbGsI5To1doX/MZ3J6bCyfsTlPrpWwV8euYXpEENyg+Cnmgep3h8Y0TGmU3wN0xTBxNR5bGgRmYDor6TgzLf+3Sk92bQfLOZcYFMvY5NkwV9r+PAh6UMokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$4\\cdot\\:\\frac{1}{4}\\ln\\left|x+1\\right|=\\ln\\left|x+1\\right|$$", "input": "4\\cdot\\:\\frac{1}{4}\\ln\\left|x+1\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{4}\\ln\\left|x+1\\right|" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\ln\\left|x+1\\right|\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\ln\\left|x+1\\right|\\cdot\\:1=\\ln\\left|x+1\\right|$$", "result": "=\\ln\\left|x+1\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV8/8hEWEUvNRhNZaQUynkREQSUpL4QD/NAhVAQA6X0x12ICf2WQN9mSJxQaQ7cQX4it8wm4YsT3c+x2f2pAHISNxoOEeCrwg4+ipCfx3vvrVpHRHdzS5g2HJFanH3nVRLiHiXj0Ip1doXRPIJCQFOu2Ty86nXYS3TGYGARqirajlQjoZYr/6mXnYDx8qKaA8+zJzgy5lQYdHYnhRCAmsaJj6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bRuMNnk0kgWe4IBgNnOroM4/CtXhQgVmPBs4zSzq+3pemPzwNZOmrxuVX8fmdo7AxLqf3xB7UKGUnLLcaMomsqvrn1EtKakWevwqRvaHn/9vjYWmLCPfXiY4awC5ECN6ICf2WQN9mSJxQaQ7cQX4it8wm4YsT3c+x2f2pAHISNwceIub5thG0f3YiHWURVQ+2xIDMLixv81/pLbwYP8U0ZTeMz0beO/zP42eKH86ENmxJKtZDbQXQwwZdrZKNHdAixpjZyz1hn/Tx95aZo6CSIArIGGfg6EjduQ00UGxnhT/IRFhFLzUYTWWkFMp5EREhukgpuafdbdotybI6d6+MQ2TlG3ZjZ/r9SsU9Mg8K6dhuGFrgnoTUmHtQOeiDBkGzxtf7vSy95fP1uFgbauID3XB6CXd1z/1s4GGziBvyWY=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\ln\\left|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\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|x-1\\right|-\\frac{2}{x-1}-\\ln\\left|x+1\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }