integral of 8xln(7x)

{ "query": { "display": "$$\\int\\:8x\\ln\\left(7x\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int 8x\\ln(7x)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "8(\\frac{1}{2}x^{2}\\ln(7x)-\\frac{x^{2}}{4})+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:8x\\ln\\left(7x\\right)dx=8\\left(\\frac{1}{2}x^{2}\\ln\\left(7x\\right)-\\frac{x^{2}}{4}\\right)+C$$", "input": "\\int\\:8x\\ln\\left(7x\\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\\:x\\ln\\left(7x\\right)dx" }, { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:x\\ln\\left(7x\\right)dx", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=\\ln\\left(7x\\right)$$" }, { "type": "step", "primary": "$$v'=x$$" }, { "type": "interim", "title": "$$u'=\\frac{d}{dx}\\left(\\ln\\left(7x\\right)\\right)=\\frac{1}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(7x\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{7x}\\frac{d}{dx}\\left(7x\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(7x\\right)\\right)", "result": "=\\frac{1}{7x}\\frac{d}{dx}\\left(7x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\ln\\left(u\\right),\\:\\:u=7x$$" ], "result": "=\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)\\frac{d}{dx}\\left(7x\\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(7x\\right)" }, { "type": "step", "primary": "Substitute back $$u=7x$$", "result": "=\\frac{1}{7x}\\frac{d}{dx}\\left(7x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs4JBa7+xAxYk5IUoNXi/PT8zeERICEnv1Ds5A1/BdIw2AW2hhsmfO/7M0RMNsJBcxhgKq+X573ilbTfYIBcNAzWM9HA0a5VHxu5rWRbGwYV40q3NE14GqI77URpJaNaT4lozz2S4yBUvBut+wm++fsEuDOVaQvKofqHoY5jNapszx1SRmkdwriKeTV96yww0ImpXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(7x\\right)=7$$", "input": "\\frac{d}{dx}\\left(7x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=7\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=7\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=7", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp7qFWeq0o6NNhqfp4hNnxLZGku9zFkxwe1dTH8vycb9ynYo8KSgQC8g4nDI37qae1NbbqpyK7JQEZdATEJR51jattTSeH8LNnjYKVph2VwY" } }, { "type": "step", "result": "=\\frac{1}{7x}\\cdot\\:7" }, { "type": "interim", "title": "Simplify $$\\frac{1}{7x}\\cdot\\:7:{\\quad}\\frac{1}{x}$$", "input": "\\frac{1}{7x}\\cdot\\:7", "result": "=\\frac{1}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:7}{7x}" }, { "type": "step", "primary": "Cancel the common factor: $$7$$", "result": "=\\frac{1}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Dcr4Cc5ECUAQRn2hHxz7L7Ez17UsZSD0BsOHbW27DC8JQJZuTAY5js+oqjdT8kslr478qsqVqgwLmYYu6zfToT/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10L21/JBqiaM9prSIlShMA7T6x+kewy002Gpl+2X/NCNk=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$v=\\int\\:xdx=\\frac{x^{2}}{2}$$", "input": "\\int\\:xdx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=\\frac{x^{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{1+1}}{1+1}:{\\quad}\\frac{x^{2}}{2}$$", "input": "\\frac{x^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7814/6/Jz6acDoAMznrJ9GL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpPzIcDl+e6/8g9uDsiVdOq//YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{x^{2}}{2}+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(7x\\right)\\frac{x^{2}}{2}-\\int\\:\\frac{1}{x}\\cdot\\:\\frac{x^{2}}{2}dx" }, { "type": "interim", "title": "Simplify", "input": "\\ln\\left(7x\\right)\\frac{x^{2}}{2}-\\int\\:\\frac{1}{x}\\cdot\\:\\frac{x^{2}}{2}dx", "result": "=\\frac{1}{2}x^{2}\\ln\\left(7x\\right)-\\int\\:\\frac{x}{2}dx", "steps": [ { "type": "interim", "title": "Multiply $$\\ln\\left(7x\\right)\\frac{x^{2}}{2}\\::{\\quad}\\frac{x^{2}\\ln\\left(7x\\right)}{2}$$", "input": "\\ln\\left(7x\\right)\\frac{x^{2}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x^{2}\\ln\\left(7x\\right)}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{2}\\ln\\left(7x\\right)}{2}-\\int\\:\\frac{1}{x}\\cdot\\:\\frac{x^{2}}{2}dx" }, { "type": "interim", "title": "Multiply $$\\frac{1}{x}\\cdot\\:\\frac{x^{2}}{2}\\::{\\quad}\\frac{x}{2}$$", "input": "\\frac{1}{x}\\cdot\\:\\frac{x^{2}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:x^{2}}{x\\cdot\\:2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=\\frac{x^{2}}{2x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{x}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{2}\\ln\\left(7x\\right)}{2}-\\int\\:\\frac{x}{2}dx" }, { "type": "step", "result": "=\\frac{1}{2}x^{2}\\ln\\left(7x\\right)-\\int\\:\\frac{x}{2}dx" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s72/Gvmg9JZ+apo/cmIwRe1QsjvX7KVUO/AeCFSId4S33E0tNgVOMmbzw/FtncwXm1usNhL3/rViSpTA6uTk6QaTWM9HA0a5VHxu5rWRbGwYVOHAv0KXl7jZTliGE42d04ritWVNYd4cfteWD/YmvLvnvxAfZ1n81s9fbWlp23h7XBLgzlWkLyqH6h6GOYzWqbBNLTYFTjJm88PxbZ3MF5tYHvBbCwv/grBO+i5nI5gly" } }, { "type": "step", "result": "=8\\left(\\frac{1}{2}x^{2}\\ln\\left(7x\\right)-\\int\\:\\frac{x}{2}dx\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{x}{2}dx=\\frac{x^{2}}{4}$$", "input": "\\int\\:\\frac{x}{2}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{2}\\cdot\\:\\int\\:xdx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=\\frac{1}{2}\\cdot\\:\\frac{x^{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{1+1}}{1+1}:{\\quad}\\frac{x^{2}}{2}$$", "input": "\\frac{x^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7814/6/Jz6acDoAMznrJ9GL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpPzIcDl+e6/8g9uDsiVdOq//YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\cdot\\:\\frac{x^{2}}{2}:{\\quad}\\frac{x^{2}}{4}$$", "input": "\\frac{1}{2}\\cdot\\:\\frac{x^{2}}{2}", "result": "=\\frac{x^{2}}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:x^{2}}{2\\cdot\\:2}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{x^{2}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8f/5NfX25yNOq3n2FMxADOuYcM2VFKV06omE0l+DfOFfcJChiVhDxT5N/LHSTLMjyL3fsR+LfX/oMdD1Fu/8vudxNmeb/xhatr/XL8I0nxNMHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2rseciYP0aoDoOR8D6Ko6BTC2oCuEiBa/QWHm2Q4KKov9A==" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=8\\left(\\frac{1}{2}x^{2}\\ln\\left(7x\\right)-\\frac{x^{2}}{4}\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=8\\left(\\frac{1}{2}x^{2}\\ln\\left(7x\\right)-\\frac{x^{2}}{4}\\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=Integration%20By%20Parts", "practiceTopic": "Integration by Parts" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=8(\\frac{1}{2}x^{2}\\ln(7x)-\\frac{x^{2}}{4})+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }