integral of xe^xln(x-1)

{ "query": { "display": "$$\\int\\:xe^{x}\\ln\\left(x-1\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int xe^{x}\\ln(x-1)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\ln(x-1)(e^{x}x-e^{x})-e^{x}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:xe^{x}\\ln\\left(x-1\\right)dx=\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-e^{x}+C$$", "input": "\\int\\:xe^{x}\\ln\\left(x-1\\right)dx", "steps": [ { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:xe^{x}\\ln\\left(x-1\\right)dx", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=\\ln\\left(x-1\\right)$$" }, { "type": "step", "primary": "$$v'=e^{x}x$$" }, { "type": "interim", "title": "$$u'=\\frac{d}{dx}\\left(\\ln\\left(x-1\\right)\\right)=\\frac{1}{x-1}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x-1\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{x-1}\\frac{d}{dx}\\left(x-1\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x-1\\right)\\right)", "result": "=\\frac{1}{x-1}\\frac{d}{dx}\\left(x-1\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\ln\\left(u\\right),\\:\\:u=x-1$$" ], "result": "=\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)\\frac{d}{dx}\\left(x-1\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "input": "\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "result": "=\\frac{1}{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoqTCAmruKWcJsn66ZPDMT8cjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJq1PlbV5jLoKv9solFCc4blTW26qciuyUBGXQExCUedYd9mDo5FIvzrirtH7/W8pPUxk6YPA4jUd3Af4X0JJJ64=" } }, { "type": "step", "result": "=\\frac{1}{u}\\frac{d}{dx}\\left(x-1\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-1$$", "result": "=\\frac{1}{x-1}\\frac{d}{dx}\\left(x-1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYq2LYTlWBtDFmxOV4DO8nkiQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEyDWCSqwfKB+LJy+pyziHhwBVcUz+pAPrcpqIfBB8VlawGZb9REnJZng7AuxkmDlRiqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x-1\\right)=1$$", "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", "result": "=\\frac{1}{x-1}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{x-1}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$v=\\int\\:e^{x}xdx=e^{x}x-e^{x}$$", "input": "\\int\\:e^{x}xdx", "steps": [ { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:e^{x}xdx", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=x$$" }, { "type": "step", "primary": "$$v'=e^{x}$$" }, { "type": "interim", "title": "$$u'=\\frac{dx}{dx}=1$$", "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+8ZDu8iF4MSewt4yms1lIaSUX15NRU/Cc5dZXPCJmBAwLps2lra35+1iz7ujEYdV" } }, { "type": "interim", "title": "$$v=\\int\\:e^{x}dx=e^{x}$$", "input": "\\int\\:e^{x}dx", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{x}dx=e^{x}$$", "result": "=e^{x}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=e^{x}+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", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7yCokk4TpRsHPBkcvvr53At57+9tLAebyt/n2mi8F8LilkhUx2zhdhO4nW7WzpnEuXjgtPDRD2GNV8rR5eC2BiDP7QNZoPVbFc3iAKhbic4hQHyKCEdC0LUiJ7zm+Nz3iA==" } }, { "type": "step", "result": "=xe^{x}-\\int\\:1\\cdot\\:e^{x}dx" }, { "type": "interim", "title": "Simplify", "input": "xe^{x}-\\int\\:1\\cdot\\:e^{x}dx", "result": "=e^{x}x-\\int\\:e^{x}dx", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{x}=e^{x}$$", "result": "=e^{x}x-\\int\\:e^{x}dx" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xdQC0d/F9CWsCOcmSDaJY+FjYEyrvCWvj0UgozEWLeQeSQilmHh9Rr5YjxdBgITC8051+98xgjCy9hiyp0OchvVSqG07GELhkHBYDWC+OCbFhQjN5tlIxb4hTGF5r1Oq6qQhfqSx4Ip0S2H5wgufDI0B4tKj1rjo3lSvhuHyFWXtgUxf6TtJKhZ0O9mS/RSDw==" } }, { "type": "step", "result": "=e^{x}x-\\int\\:e^{x}dx" }, { "type": "interim", "title": "$$\\int\\:e^{x}dx=e^{x}$$", "input": "\\int\\:e^{x}dx", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{x}dx=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7yCokk4TpRsHPBkcvvr53At57+9tLAebyt/n2mi8F8LilkhUx2zhdhO4nW7WzpnEuVroeUCC5gNxQc9h7CboxnOdNk9NfqKD2n0pqiLxXpUzuzoDhpIcHmn9MxBWmjZQyQ==" } }, { "type": "step", "result": "=e^{x}x-e^{x}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=e^{x}x-e^{x}+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", "interimType": "Integrals" } }, { "type": "step", "result": "=\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-\\int\\:\\frac{1}{x-1}\\left(e^{x}x-e^{x}\\right)dx" }, { "type": "interim", "title": "Simplify", "input": "\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-\\int\\:\\frac{1}{x-1}\\left(e^{x}x-e^{x}\\right)dx", "result": "=\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-\\int\\:e^{x}dx", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{1}{x-1}\\left(e^{x}x-e^{x}\\right)\\::{\\quad}e^{x}$$", "input": "\\frac{1}{x-1}\\left(e^{x}x-e^{x}\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\left(e^{x}x-e^{x}\\right)}{x-1}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(e^{x}x-e^{x}\\right)=e^{x}x-e^{x}$$", "input": "1\\cdot\\:\\left(e^{x}x-e^{x}\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(e^{x}x-e^{x}\\right)=\\left(e^{x}x-e^{x}\\right)$$", "result": "=\\left(e^{x}x-e^{x}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=e^{x}x-e^{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UVZJzou816Rf+2CDVEOACA+I2W8HSFa8naL34TyRYBJ1g99dC9fj9sg0EHzBIRDRds+/tc92JsOUygsMOneTU/8//6/nV5O4fb8Xgwi7map9K1BxdsQWHhjHJGT7Q3j/rFPYgDrdK5pEL4Wvb7GRhHW6h/Kz9RWwAbb7HdW6+w0=" } }, { "type": "step", "result": "=\\frac{e^{x}x-e^{x}}{x-1}" }, { "type": "step", "primary": "Factor out common term $$e^{x}$$", "result": "=\\frac{e^{x}\\left(x-1\\right)}{x-1}", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "primary": "Cancel the common factor: $$x-1$$", "result": "=e^{x}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-\\int\\:e^{x}dx" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77iG5rzKeZcj1HILgxswYGAQtR4odGyFfP+b9O40fYTMFqPJweNQLET69s1yBMlEJTliFkIunLQICXzsisHSh9fnXjtJUe8XnWySWceuXEv7fKKN+u0L4hpSMWfwiXjyjs+8nLdwp5vTjbmK0KrBG9uQ1Xa7hVHArl3wlOlTyCDjkxv+8AweQrL/3huZq2oQdHkkIpZh4fUa+WI8XQYCEwulU/+xj9Dc1qUVFGqGPhoY" } }, { "type": "step", "result": "=\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-\\int\\:e^{x}dx" }, { "type": "interim", "title": "$$\\int\\:e^{x}dx=e^{x}$$", "input": "\\int\\:e^{x}dx", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{x}dx=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7yCokk4TpRsHPBkcvvr53At57+9tLAebyt/n2mi8F8LilkhUx2zhdhO4nW7WzpnEuVroeUCC5gNxQc9h7CboxnOdNk9NfqKD2n0pqiLxXpUzuzoDhpIcHmn9MxBWmjZQyQ==" } }, { "type": "step", "result": "=\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-e^{x}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\ln\\left(x-1\\right)\\left(e^{x}x-e^{x}\\right)-e^{x}+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=Integration%20By%20Parts", "practiceTopic": "Integration by Parts" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\ln(x-1)(e^{x}x-e^{x})-e^{x}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }