integral of (11)/(11+e^x)

{ "query": { "display": "$$\\int\\:\\frac{11}{11+e^{x}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{11}{11+e^{x}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "x-\\ln\\left|e^{x}+11\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{11}{11+e^{x}}dx=x-\\ln\\left|e^{x}+11\\right|+C$$", "input": "\\int\\:\\frac{11}{11+e^{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": "=11\\cdot\\:\\int\\:\\frac{1}{11+e^{x}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{11+e^{x}}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: $$u=e^{x}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7YoLr4BhtUxiRrnpz++7ljDFdbqH8rP1FbABtvsd1br7DQ==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=e^{x}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{e^{x}}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{11+u}\\cdot\\:\\frac{1}{e^{x}}du" }, { "type": "step", "primary": "$$u=e^{x}$$", "result": "=\\int\\:\\frac{1}{11+u}\\cdot\\:\\frac{1}{u}du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{11+u}\\cdot\\:\\frac{1}{u}:{\\quad}\\frac{1}{u\\left(11+u\\right)}$$", "input": "\\frac{1}{11+u}\\cdot\\:\\frac{1}{u}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{\\left(11+u\\right)u}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{u\\left(u+11\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u\\left(11+u\\right)}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s76h6o8oyw/7zNQGXANwZKPGB4TdqBy55WMrymWlw3jKzzeOY1PjgUy0LUpni0N5ZVbjaY14IHJep/ok2WQsvUcSWCGBokMAdYPyzlTTX+kUtwYZL4ioFKq2BU47KkZASboRcmQkfNhojMo7F65ZBOgIEuDOVaQvKofqHoY5jNapscN6G3wC0aNOpX3GMtOsWhImpXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "=11\\cdot\\:\\int\\:\\frac{1}{u\\left(11+u\\right)}du" }, { "type": "interim", "title": "Take the partial fraction of $$\\frac{1}{u\\left(11+u\\right)}:{\\quad}\\frac{1}{11u}-\\frac{1}{11\\left(u+11\\right)}$$", "input": "\\frac{1}{u\\left(11+u\\right)}", "steps": [ { "type": "interim", "title": "Create the partial fraction template using the denominator $$u\\left(u+11\\right)$$", "result": "\\frac{1}{u\\left(u+11\\right)}=\\frac{a_{0}}{u}+\\frac{a_{1}}{u+11}", "steps": [ { "type": "step", "primary": "For $$u\\:$$add the partial fraction(s): $$\\frac{a_{0}}{u}$$" }, { "type": "step", "primary": "For $$u+11\\:$$add the partial fraction(s): $$\\frac{a_{1}}{u+11}$$" }, { "type": "step", "result": "\\frac{1}{u\\left(u+11\\right)}=\\frac{a_{0}}{u}+\\frac{a_{1}}{u+11}" } ], "meta": { "interimType": "Partial Fraction Templates Top 1Eq" } }, { "type": "step", "primary": "Multiply equation by the denominator", "result": "\\frac{1\\cdot\\:u\\left(u+11\\right)}{u\\left(u+11\\right)}=\\frac{a_{0}u\\left(u+11\\right)}{u}+\\frac{a_{1}u\\left(u+11\\right)}{u+11}" }, { "type": "step", "primary": "Simplify", "result": "1=a_{0}\\left(u+11\\right)+a_{1}u" }, { "type": "step", "primary": "Solve the unknown parameters by plugging the real roots of the denominator: $$0,\\:-11$$" }, { "type": "interim", "title": "For the denominator root $$0:{\\quad}a_{0}=\\frac{1}{11}$$", "steps": [ { "type": "step", "primary": "Plug in $$x=0\\:$$into the equation", "result": "1=a_{0}\\left(0+11\\right)+a_{1}\\cdot\\:0" }, { "type": "step", "primary": "Expand", "result": "1=11a_{0}" }, { "type": "interim", "title": "Solve $$1=11a_{0}\\:$$for $$a_{0}:{\\quad}a_{0}=\\frac{1}{11}$$", "input": "1=11a_{0}", "result": "a_{0}=\\frac{1}{11}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "11a_{0}=1" }, { "type": "interim", "title": "Divide both sides by $$11$$", "input": "11a_{0}=1", "result": "a_{0}=\\frac{1}{11}", "steps": [ { "type": "step", "primary": "Divide both sides by $$11$$", "result": "\\frac{11a_{0}}{11}=\\frac{1}{11}" }, { "type": "step", "primary": "Simplify", "result": "a_{0}=\\frac{1}{11}" } ], "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 $$-11:{\\quad}a_{1}=-\\frac{1}{11}$$", "steps": [ { "type": "step", "primary": "Plug in $$x=-11\\:$$into the equation", "result": "1=a_{0}\\left(\\left(-11\\right)+11\\right)+a_{1}\\left(-11\\right)" }, { "type": "step", "primary": "Expand", "result": "1=-11a_{1}" }, { "type": "interim", "title": "Solve $$1=-11a_{1}\\:$$for $$a_{1}:{\\quad}a_{1}=-\\frac{1}{11}$$", "input": "1=-11a_{1}", "result": "a_{1}=-\\frac{1}{11}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-11a_{1}=1" }, { "type": "interim", "title": "Divide both sides by $$-11$$", "input": "-11a_{1}=1", "result": "a_{1}=-\\frac{1}{11}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-11$$", "result": "\\frac{-11a_{1}}{-11}=\\frac{1}{-11}" }, { "type": "step", "primary": "Simplify", "result": "a_{1}=-\\frac{1}{11}" } ], "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}{11},\\:a_{1}=-\\frac{1}{11}" }, { "type": "step", "primary": "Plug the solutions to the partial fraction parameters to obtain the final result", "result": "\\frac{\\frac{1}{11}}{u}+\\frac{\\left(-\\frac{1}{11}\\right)}{u+11}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\frac{1}{11}}{u}+\\frac{\\left(-\\frac{1}{11}\\right)}{u+11}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{11}}{u}:{\\quad}\\frac{1}{11u}$$", "input": "\\frac{\\frac{1}{11}}{u}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{11u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88aja7wq/xBonYS//5h+VfyFTTNGoPE9TME3q+OPmgkv2RQiEw6G4T+RFI2ZfZDoB3kMp4rWQmabJIPe/sw67xsb+2LGmNnLPWGf9PH3lpmjoJIT6Wk1j0nJ3Nx9hn2Bb6EBnQFNO5Y8SuIqbmMv7gNm3qwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Simplify $$\\frac{\\left(-\\frac{1}{11}\\right)}{u+11}:{\\quad}-\\frac{1}{11\\left(u+11\\right)}$$", "input": "\\frac{-\\frac{1}{11}}{u+11}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{1}{11}}{u+11}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{1}{11}}{u+11}=\\frac{1}{11\\left(u+11\\right)}$$" ], "result": "=-\\frac{1}{11\\left(u+11\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVf6ag2pBpoUHPrUmPV4i7nxV00rpv8+ZC6TM10tVCSHshgFkMSyGhkcJufVqsMeq9FwbbztTHTZLU+AhkI+ApX9kS3dlcCKpQTQcheuut7MkdkqXaZXgCXO2IClMAWKDPelZEZk4JDvk+Rt7TOxGLKFBjolEStWsMlSe37r28bEFsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=\\frac{1}{11u}-\\frac{1}{11\\left(u+11\\right)}" } ], "meta": { "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\frac{1}{11u}-\\frac{1}{11\\left(u+11\\right)}" } ], "meta": { "solvingClass": "Partial Fractions", "interimType": "Algebraic Manipulation Partial Fraction Top Title 1Eq" } }, { "type": "step", "result": "=11\\cdot\\:\\int\\:\\frac{1}{11u}-\\frac{1}{11\\left(u+11\\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": "=11\\left(\\int\\:\\frac{1}{11u}du-\\int\\:\\frac{1}{11\\left(u+11\\right)}du\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{11u}du=\\frac{1}{11}\\ln\\left|u\\right|$$", "input": "\\int\\:\\frac{1}{11u}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}{11}\\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}{11}\\ln\\left|u\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{11\\left(u+11\\right)}du=\\frac{1}{11}\\ln\\left|u+11\\right|$$", "input": "\\int\\:\\frac{1}{11\\left(u+11\\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}{11}\\cdot\\:\\int\\:\\frac{1}{u+11}du" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{u+11}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: $$v=u+11$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=1$$", "input": "\\frac{d}{du}\\left(u+11\\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(11\\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": "$$\\frac{d}{du}\\left(11\\right)=0$$", "input": "\\frac{d}{du}\\left(11\\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+idP5vLVrjUll6eMdYnpKWjncVObGB7N+7S9KhADZGku9zFkxwe1dTH8vycb9TbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51i7VnujxF5qCVWJHZWt5+OF" } }, { "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+4/5L7s7+XcmPIK75L5hEJ/NPkosnEcFWhAQxlpHiQtWiL6gUa07CmuVIP0/DMlFt6wE9n+ApCJDtyZjhH1bmxVOkEAjY9vjXyBQRTX7G9tDG601GzT/qOmQR2x/5vkO5CnG+Sl89bA+zX4bD3u3gx65o2NJhOP8jPOjUAODCISjG3EPUUsKIcdyTU2qbTbjVtrM8zqhQ==" } }, { "type": "step", "result": "=\\frac{1}{11}\\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}{11}\\ln\\left|v\\right|" }, { "type": "step", "primary": "Substitute back $$v=u+11$$", "result": "=\\frac{1}{11}\\ln\\left|u+11\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=11\\left(\\frac{1}{11}\\ln\\left|u\\right|-\\frac{1}{11}\\ln\\left|u+11\\right|\\right)" }, { "type": "step", "primary": "Substitute back $$u=e^{x}$$", "result": "=11\\left(\\frac{1}{11}\\ln\\left|e^{x}\\right|-\\frac{1}{11}\\ln\\left|e^{x}+11\\right|\\right)" }, { "type": "interim", "title": "Simplify $$11\\left(\\frac{1}{11}\\ln\\left|e^{x}\\right|-\\frac{1}{11}\\ln\\left|e^{x}+11\\right|\\right):{\\quad}x-\\ln\\left|e^{x}+11\\right|$$", "input": "11\\left(\\frac{1}{11}\\ln\\left|e^{x}\\right|-\\frac{1}{11}\\ln\\left|e^{x}+11\\right|\\right)", "result": "=x-\\ln\\left|e^{x}+11\\right|", "steps": [ { "type": "interim", "title": "$$\\frac{1}{11}\\ln\\left|e^{x}\\right|=\\frac{1}{11}x$$", "input": "\\frac{1}{11}\\ln\\left|e^{x}\\right|", "steps": [ { "type": "interim", "title": "Simplify $$\\left|e^{x}\\right|:{\\quad}e^{x}$$", "input": "\\left|e^{x}\\right|", "result": "=\\frac{1}{11}\\ln\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply absolute rule: $$a>0,\\:\\left|a^{x}\\right|=a^{x}$$", "secondary": [ "$$\\left|e^{x}\\right|=e^{x}$$" ], "result": "=e^{x}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\ln\\left(e^{x}\\right):{\\quad}x$$", "input": "\\ln\\left(e^{x}\\right)", "result": "=\\frac{1}{11}x", "steps": [ { "type": "step", "primary": "Apply log rule $$\\log_{a}\\left(x^b\\right)=b\\cdot\\log_{a}\\left(x\\right),\\:\\quad$$ assuming $$x\\:\\geq\\:0$$", "result": "=\\ln\\left(e\\right)x" }, { "type": "step", "primary": "Apply log rule: $$\\log_a\\left(a\\right)=1$$", "result": "=x", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+shdDvdLjGglGcDraL+zSbNOrjAxrU3mTCZ5Et5didnuuq/ABrW1gYcDNePOT4ahCUCWbkwGOY7PqKo3U/JLJbcA2hHP7Gp2rsnHCG/QziKtP/D7TqR2jRE/NuDAcTmMiHBhlfkpI7IY1DzLNxccZbNOrjAxrU3mTCZ5Et5didmwn6Romj5V2CkEhkUbQxKLiiqHaRVkr1+qAJvi3xGSDg==" } }, { "type": "step", "result": "=11\\left(\\frac{1}{11}x-\\frac{1}{11}\\ln\\left|e^{x}+11\\right|\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=11,\\:b=\\frac{1}{11}x,\\:c=\\frac{1}{11}\\ln\\left|e^{x}+11\\right|$$" ], "result": "=11\\cdot\\:\\frac{1}{11}x-11\\cdot\\:\\frac{1}{11}\\ln\\left|e^{x}+11\\right|", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$11\\cdot\\:\\frac{1}{11}x-11\\cdot\\:\\frac{1}{11}\\ln\\left|e^{x}+11\\right|:{\\quad}x-\\ln\\left|e^{x}+11\\right|$$", "input": "11\\cdot\\:\\frac{1}{11}x-11\\cdot\\:\\frac{1}{11}\\ln\\left|e^{x}+11\\right|", "result": "=x-\\ln\\left|e^{x}+11\\right|", "steps": [ { "type": "interim", "title": "$$11\\cdot\\:\\frac{1}{11}x=x$$", "input": "11\\cdot\\:\\frac{1}{11}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:11}{11}x" }, { "type": "step", "primary": "Cancel the common factor: $$11$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$x\\cdot\\:1=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HDQ7Td+D3XJM5ZO06G7zcJQvbCQC1LQH6vAq1oCr+qwDnzlbPZjyKgy1eUCFsLd5I9uWp+dc2RFudgyDeyFw7S/tqW6ogh0VKwg9cZ/7zXo8qutmNvHtXiwvR39Vr/DDsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$11\\cdot\\:\\frac{1}{11}\\ln\\left|e^{x}+11\\right|=\\ln\\left|e^{x}+11\\right|$$", "input": "11\\cdot\\:\\frac{1}{11}\\ln\\left|e^{x}+11\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:11}{11}\\ln\\left|e^{x}+11\\right|" }, { "type": "step", "primary": "Cancel the common factor: $$11$$", "result": "=\\ln\\left|e^{x}+11\\right|\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\ln\\left|e^{x}+11\\right|\\cdot\\:1=\\ln\\left|e^{x}+11\\right|$$", "result": "=\\ln\\left|e^{x}+11\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HDQ7Td+D3XJM5ZO06G7zcO0wCUwujdzWrj3yopQjI0cD1PEsIHo5KS7HJSGCdzTdXo9WRZzG4ildXirIMe/hjqORWLXkjysF58uLgjK3bCsbwlV7T205OcXO74h6eJZ1YD6wxtmNqrUdq4vq+veW2VA6McQe+5dBcu8kesD4LtxAHPf2zJbapyfV0fx1DEByFsC2wuu/C3mfRpIVVlgxrrwff/AlEiEj/Y4TJVu4svWamkuMrf55Go53mdz2LEc9StZwhe4OBOp48Wetr6JU5g==" } }, { "type": "step", "result": "=x-\\ln\\left|e^{x}+11\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+pkcywRqLRDm6/NpZIqzaCo+OCi6ZF8MODHcSMLaP897x4m8wAF6NWCG9WTs8VC97TAJTC6N3NauPfKilCMjRwPU8SwgejkpLsclIYJ3NN0SkWgxyS5xDRRIcZ68mOBMzMFYmi1F5Hg/ibpEToVnYwbTboB4+h0qlrHUQFque5yR8+jUSzTrnroiMDTxp1yco3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCAf0TfRC22eRcHJcmEP73Q9FsC2wuu/C3mfRpIVVlgxrgsJzystfwjVqHOeL4CsSlC1i8K74z6sWzEwDmStZdBu3rNvl8qf5iTDO7apXzt9zq1AooF4/c7la/jTXLsSqcs=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=x-\\ln\\left|e^{x}+11\\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=x-\\ln\\left|e^{x}+11\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }