integral of (3-1/x)^{-2}(1/(x^2))

{ "query": { "display": "$$\\int\\:\\left(3-\\frac{1}{x}\\right)^{-2}\\left(\\frac{1}{x^{2}}\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int (3-\\frac{1}{x})^{-2}(\\frac{1}{x^{2}})dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{x}{(3x-1)^{2}}+\\frac{1}{3(3x-1)^{2}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\left(3-\\frac{1}{x}\\right)^{-2}\\left(\\frac{1}{x^{2}}\\right)dx=-\\frac{x}{\\left(3x-1\\right)^{2}}+\\frac{1}{3\\left(3x-1\\right)^{2}}+C$$", "input": "\\int\\:\\left(3-\\frac{1}{x}\\right)^{-2}\\frac{1}{x^{2}}dx", "steps": [ { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:\\left(3-\\frac{1}{x}\\right)^{-2}\\frac{1}{x^{2}}dx", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=\\left(3-\\frac{1}{x}\\right)^{-2}$$" }, { "type": "step", "primary": "$$v'=\\frac{1}{x^{2}}$$" }, { "type": "interim", "title": "$$u'=\\frac{d}{dx}\\left(\\left(3-\\frac{1}{x}\\right)^{-2}\\right)=-\\frac{2x}{\\left(3x-1\\right)^{3}}$$", "input": "\\frac{d}{dx}\\left(\\left(3-\\frac{1}{x}\\right)^{-2}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}-\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}\\frac{d}{dx}\\left(3-\\frac{1}{x}\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(3-\\frac{1}{x}\\right)^{-2}\\right)", "result": "=-\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}\\frac{d}{dx}\\left(3-\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{-2},\\:\\:u=\\left(3-\\frac{1}{x}\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{-2}\\right)\\frac{d}{dx}\\left(\\left(3-\\frac{1}{x}\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{-2}\\right)=-\\frac{2}{u^{3}}$$", "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": "interim", "title": "Simplify $$-2u^{-2-1}:{\\quad}-\\frac{2}{u^{3}}$$", "input": "-2u^{-2-1}", "result": "=-\\frac{2}{u^{3}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-2-1=-3$$", "result": "=-2u^{-3}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-3}=\\frac{1}{u^{3}}$$" ], "result": "=2\\cdot\\:\\frac{1}{u^{3}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:2}{u^{3}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-\\frac{2}{u^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GBuGcf4FRLkDrClwSBFsKSAn9lkDfZkicUGkO3EF+IqZaJMSlCNq3dI7v+XabnyXBdbd0nY7/2v0KjeKDrfjY/C30sSftAIFS6Qkpy19Ikq4k8jnENbqCWh0r36iEosf8M4UjNpr4+T1YRXnCpEOsw==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\frac{2}{u^{3}}\\frac{d}{dx}\\left(\\left(3-\\frac{1}{x}\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\left(3-\\frac{1}{x}\\right)$$", "result": "=-\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}\\frac{d}{dx}\\left(3-\\frac{1}{x}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp7zGuKXoNDi2+4UAdrA4nPjILTLEkhc3G5yA9FDbHM/pN4cZPWgnwFqHQUcV4FHseJOklaVyfAA4+IQ2/V91lnjgT/SrQIc7mFzkmCezRK3zlCrbzDSNsvL6pECYZb8uPiw0V8XsVVRA6nkEeBV6vo7IaJXOwYeDmc7EzljHTwJX4fU4M0rvAsQbp8oPGGq02RLd2VwIqlBNByF6663sySqRIrj0nXsAzFXrI66N/96pMf4devEp9a7DxWbDdVagA==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3-\\frac{1}{x}\\right)=\\frac{1}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(3-\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(3\\right)-\\frac{d}{dx}\\left(\\frac{1}{x}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3\\right)=0$$", "input": "\\frac{d}{dx}\\left(3\\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+idP5vLVrjUll6eMdYu6nPER/cBcxgb/Kz63vQV1J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTuwXg0Wd+I5tymlezl5JoPF" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{1}{x}\\right)=-\\frac{1}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=\\frac{d}{dx}\\left(x^{-1}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=-1\\cdot\\:x^{-1-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$-1\\cdot\\:x^{-1-1}:{\\quad}-\\frac{1}{x^{2}}$$", "input": "-1\\cdot\\:x^{-1-1}", "result": "=-\\frac{1}{x^{2}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-1\\cdot\\:x^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-2}=\\frac{1}{x^{2}}$$" ], "result": "=-1\\cdot\\:\\frac{1}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{2}}=\\frac{1}{x^{2}}$$", "result": "=-\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+AgNZzeIicTfbr51/JGq1KiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrT1r2iaXj0z6hiHFOta70Gf8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUl/dfOQfCkILEECUyPIy9DrRF1+E4wvPRIGnJs5KwUnrw=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=0-\\left(-\\frac{1}{x^{2}}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{x^{2}}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}\\cdot\\:\\frac{1}{x^{2}}" }, { "type": "interim", "title": "Simplify $$-\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}\\cdot\\:\\frac{1}{x^{2}}:{\\quad}-\\frac{2x}{\\left(3x-1\\right)^{3}}$$", "input": "-\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}\\cdot\\:\\frac{1}{x^{2}}", "result": "=-\\frac{2x}{\\left(3x-1\\right)^{3}}", "steps": [ { "type": "interim", "title": "$$\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}=\\frac{2x^{3}}{\\left(3x-1\\right)^{3}}$$", "input": "\\frac{2}{\\left(3-\\frac{1}{x}\\right)^{3}}", "steps": [ { "type": "interim", "title": "$$\\left(3-\\frac{1}{x}\\right)^{3}=\\frac{\\left(3x-1\\right)^{3}}{x^{3}}$$", "input": "\\left(3-\\frac{1}{x}\\right)^{3}", "steps": [ { "type": "interim", "title": "Join $$3-\\frac{1}{x}:{\\quad}\\frac{3x-1}{x}$$", "input": "3-\\frac{1}{x}", "result": "=\\left(\\frac{3x-1}{x}\\right)^{3}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3x}{x}$$", "result": "=\\frac{3x}{x}-\\frac{1}{x}" }, { "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{3x-1}{x}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(3x-1\\right)^{3}}{x^{3}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JgfZrq7C5QENYDeYHXcFTkaIvJ4QhnSmXwvebxAyoVirju+5Z51e/ZZSD3gRHwjB4TnCQQt04gjxohGMK9QXomp15qPOn6XUfJLZbLYdknlGVj8+cts/p72TekmIjdXreJOxB8ClbfdcLEwS8rzmoksRYZFFglaIh0isziYUmm0trTqyZGsvP7k4Z4DhuPc+" } }, { "type": "step", "result": "=\\frac{2}{\\frac{\\left(3x-1\\right)^{3}}{x^{3}}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{2x^{3}}{\\left(3x-1\\right)^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KHFUDpDomxiAJy6usVifWniTsQfApW33XCxMEvK85qJ8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcaupfNWAR8Ox2rHq1ZCfuLeBnm4DMV5N2KurbVOhhy0gt4SJCVzy8wKgqZjxPJC0p8IoJYRKVdLtLd5nLrMtFHxFa++eFZMX/zX/H8EMJSsaTFlp8ZUqdlYe/Gj57e2SAIKhZFQ8OchBdgxMOJrEMyk3s=" } }, { "type": "step", "result": "=-\\frac{2x^{3}}{\\left(3x-1\\right)^{3}}\\cdot\\:\\frac{1}{x^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{2x^{3}\\cdot\\:1}{\\left(3x-1\\right)^{3}x^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=-\\frac{2x^{3}}{x^{2}\\left(3x-1\\right)^{3}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{x^{3}}{x^{2}}=x^{3-2}$$" ], "result": "=\\frac{2x^{3-2}}{\\left(3x-1\\right)^{3}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$3-2=1$$", "result": "=-\\frac{2x}{\\left(3x-1\\right)^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jIEWg/JJ7b80pc/y0zkqUn8j3P8Uu5myJdtbeIv1l2vtF1gfGIXHYxzr3wxiBelAK0+JHTccw5cSmhEzAxMacwOfOVs9mPIqDLV5QIWwt3k1Y0xmsnwLI9kUcSDp+m34hFtk+2mpx77x0eDCz8B2dEUqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2Sruqpf4bmmEa6eVxFW59AJw29e1JCQis/0ejS3jElor86CV4l49CKdXaF0TyCQkBTrtk0xGF6na8Z1Ig2TP4pdcphw==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$v=\\int\\:\\frac{1}{x^{2}}dx=-\\frac{1}{x}$$", "input": "\\int\\:\\frac{1}{x^{2}}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{x^{2}}dx", "result": "=-\\frac{1}{x}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{x^{2}}=x^{-2}$$" ], "result": "=\\int\\:x^{-2}dx", "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{x^{-2+1}}{-2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{-2+1}}{-2+1}:{\\quad}-\\frac{1}{x}$$", "input": "\\frac{x^{-2+1}}{-2+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=\\frac{x^{-1}}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{x^{-1}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-x^{-1}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-1}=\\frac{1}{a}$$", "result": "=-\\frac{1}{x}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{x}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77TLhe8t2tsverUFqsrJqp+r+0zX0wqSmAVtc7NV8L0Arrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx4OQzbEnsuafNY3nX9QxDlJplQfpE8SZ3DJKH2ijIHAjwS4M5VpC8qh+oehjmM1qmweKkh+28FiXwy+Vsz8xLQiialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{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": "=\\left(3-\\frac{1}{x}\\right)^{-2}\\left(-\\frac{1}{x}\\right)-\\int\\:\\left(-\\frac{2x}{\\left(3x-1\\right)^{3}}\\right)\\left(-\\frac{1}{x}\\right)dx" }, { "type": "interim", "title": "Simplify", "input": "\\left(3-\\frac{1}{x}\\right)^{-2}\\left(-\\frac{1}{x}\\right)-\\int\\:\\left(-\\frac{2x}{\\left(3x-1\\right)^{3}}\\right)\\left(-\\frac{1}{x}\\right)dx", "result": "=-\\frac{x}{\\left(3x-1\\right)^{2}}-\\int\\:\\frac{2}{\\left(3x-1\\right)^{3}}dx", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=-\\left(3-\\frac{1}{x}\\right)^{-2}\\frac{1}{x}-\\int\\:\\frac{2x}{\\left(3x-1\\right)^{3}}\\cdot\\:\\frac{1}{x}dx" }, { "type": "interim", "title": "$$\\left(3-\\frac{1}{x}\\right)^{-2}\\frac{1}{x}=\\frac{x}{\\left(3x-1\\right)^{2}}$$", "input": "\\left(3-\\frac{1}{x}\\right)^{-2}\\frac{1}{x}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$\\left(3-\\frac{1}{x}\\right)^{-2}=\\frac{1}{\\left(3-\\frac{1}{x}\\right)^{2}}$$" ], "result": "=\\frac{1}{\\left(-\\frac{1}{x}+3\\right)^{2}}\\cdot\\:\\frac{1}{x}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "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(3-\\frac{1}{x}\\right)^{2}x}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{x\\left(-\\frac{1}{x}+3\\right)^{2}}" }, { "type": "interim", "title": "$$\\left(3-\\frac{1}{x}\\right)^{2}=\\frac{\\left(3x-1\\right)^{2}}{x^{2}}$$", "input": "\\left(3-\\frac{1}{x}\\right)^{2}", "steps": [ { "type": "interim", "title": "Join $$3-\\frac{1}{x}:{\\quad}\\frac{3x-1}{x}$$", "input": "3-\\frac{1}{x}", "result": "=\\left(\\frac{3x-1}{x}\\right)^{2}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3x}{x}$$", "result": "=\\frac{3x}{x}-\\frac{1}{x}" }, { "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{3x-1}{x}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(3x-1\\right)^{2}}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JgfZrq7C5QENYDeYHXcFTkBgnzAZkUy+vMtgHHTjQTCrju+5Z51e/ZZSD3gRHwjB4TnCQQt04gjxohGMK9QXor1SVF7Kd2geqdesSn2zRPpGVj8+cts/p72TekmIjdXrNuYtKguUiJ5JrfZVMPcWOksRYZFFglaIh0isziYUmm10tQR8+wQegjlqymrSkXm/" } }, { "type": "step", "result": "=\\frac{1}{\\frac{\\left(3x-1\\right)^{2}}{x^{2}}x}" }, { "type": "interim", "title": "Multiply $$\\frac{\\left(3x-1\\right)^{2}}{x^{2}}x\\::{\\quad}\\frac{\\left(3x-1\\right)^{2}}{x}$$", "input": "\\frac{\\left(3x-1\\right)^{2}}{x^{2}}x", "result": "=\\frac{1}{\\frac{\\left(3x-1\\right)^{2}}{x}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(3x-1\\right)^{2}x}{x^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{\\left(3x-1\\right)^{2}}{x}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{1}{\\frac{b}{c}}=\\frac{c}{b}$$", "result": "=\\frac{x}{\\left(3x-1\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JgfZrq7C5QENYDeYHXcFTjDWShSLXoOiOw3F9U+THglvdJjE5uYzqRbHKRSPD34so5FYteSPKwXny4uCMrdsK0bJmVGfBOZLqew7TcvxS/igbWcrvi2mL2rGt7RY0sDhlwJVKUhJZRipHkR0AzsWayJ2LnHBWizIqWUeCzto/ngNlc7X+nmCbj64C+/yw8lBoxcqZ8Dyy6mv8xg/84CvEA==" } }, { "type": "interim", "title": "$$\\int\\:\\frac{2x}{\\left(3x-1\\right)^{3}}\\cdot\\:\\frac{1}{x}dx=\\int\\:\\frac{2}{\\left(3x-1\\right)^{3}}dx$$", "input": "\\int\\:\\frac{2x}{\\left(3x-1\\right)^{3}}\\cdot\\:\\frac{1}{x}dx", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{2x}{\\left(3x-1\\right)^{3}}\\cdot\\:\\frac{1}{x}\\::{\\quad}\\frac{2}{\\left(3x-1\\right)^{3}}$$", "input": "\\frac{2x}{\\left(3x-1\\right)^{3}}\\cdot\\:\\frac{1}{x}", "result": "=\\int\\:\\frac{2}{\\left(3x-1\\right)^{3}}dx", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{2x\\cdot\\:1}{\\left(3x-1\\right)^{3}x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{2\\cdot\\:1}{\\left(3x-1\\right)^{3}}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{\\left(3x-1\\right)^{3}}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PtLsn6mvaGskIT+k9k+/oKuPh5zIyFzgm2ivdvMgNHrFDfyC/Jzv7uiABZ+ezuwywfAt4C+ZQsIiy8ZcIYHlGKORWLXkjysF58uLgjK3bCuFcoWCRnA6/mTsQJsSRad1fjHRl8K/qXxwLLBIpFrmfwOHJx7MiiUE6lNlUWpeMJ7M8wq9Z3hIuYUYrK0KYFK9Fs50p0vDTDT7DV4n8muqZ+BlwE1WY2rE9hn1AyVqNKWyR8qB5JdERgkHSX4w+7pP9a6DuPYnOaql/MedJnjdyQ==" } }, { "type": "step", "result": "=-\\frac{x}{\\left(3x-1\\right)^{2}}-\\int\\:\\frac{2}{\\left(3x-1\\right)^{3}}dx" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71+H1ODNK7wLEG6fKDxhqtOf8WqrixEC6btRw0+A6QDjIz/u2l0CVHDKKHubctyKe/fshZUM7VdgIrF0R9gExkA+1qmniv02iqV1S0M7IBvaDUxRA6Z8Oh1Z2ddtZRBa7ML1DpaKOSXF+8HbJi+0Mhi1pIa3MEUXaLpWV44tmhzjHrhImebhajEK3z8OnQgZ65db5xqzhVZ8xcaMDuSblaHmw1q+hjFEjKzJRgYLTRGSPrYgcdLzsnp54XEjmBcdS+9hHsuRSN5vDnManOeG5qc=" } }, { "type": "step", "result": "=-\\frac{x}{\\left(3x-1\\right)^{2}}-\\int\\:\\frac{2}{\\left(3x-1\\right)^{3}}dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{2}{\\left(3x-1\\right)^{3}}dx=-\\frac{1}{3\\left(3x-1\\right)^{2}}$$", "input": "\\int\\:\\frac{2}{\\left(3x-1\\right)^{3}}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": "=2\\cdot\\:\\int\\:\\frac{1}{\\left(3x-1\\right)^{3}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{\\left(3x-1\\right)^{3}}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: $$u=3x-1$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=3$$", "input": "\\frac{d}{dx}\\left(3x-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(3x\\right)-\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x\\right)=3$$", "input": "\\frac{d}{dx}\\left(3x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsUnAVaDXCLQOuynYB+k3fTZGku9zFkxwe1dTH8vycb9BbqPJ4kl+ElAajU+EBTcW1NbbqpyK7JQEZdATEJR51i3CwF9WgU8/+rFW242YnYD" } }, { "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": "=3-0" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=3dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{3}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{3}}\\cdot\\:\\frac{1}{3}du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{u^{3}}\\cdot\\:\\frac{1}{3}:{\\quad}\\frac{1}{3u^{3}}$$", "input": "\\frac{1}{u^{3}}\\cdot\\:\\frac{1}{3}", "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}{u^{3}\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{3u^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{3u^{3}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7wXYfiQkozYdiWLJhy6IdV5TXUoC74pNfn8Enc/ddSSgVBL8m3SUY2lAb90I4uPNB6QKCe0MV1RBYKnykv7XPQE9ub4R89baV/5kN1+S2jO9f8PLSkoWKbyPQusqP7PZfOnsuSzXUodcp+Mct7VQQSBK28PwYoM0kk3yqK7RdFxVlQDcDH2jJwG3NmckkBQuRWnFFh3GddovRvrc7ssIsew=" } }, { "type": "step", "result": "=2\\cdot\\:\\int\\:\\frac{1}{3u^{3}}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": "=2\\cdot\\:\\frac{1}{3}\\cdot\\:\\int\\:\\frac{1}{u^{3}}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{u^{3}}du", "result": "=2\\cdot\\:\\frac{1}{3}\\left(-\\frac{1}{2u^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{u^{3}}=u^{-3}$$" ], "result": "=\\int\\:u^{-3}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^{-3+1}}{-3+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{-3+1}}{-3+1}:{\\quad}-\\frac{1}{2u^{2}}$$", "input": "\\frac{u^{-3+1}}{-3+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-3+1=-2$$", "result": "=\\frac{u^{-2}}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{u^{-2}}{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-2}=\\frac{1}{u^{2}}$$" ], "result": "=-\\frac{\\frac{1}{u^{2}}}{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{1}{u^{2}}}{2}=\\frac{1}{u^{2}\\cdot\\:2}$$" ], "result": "=-\\frac{1}{u^{2}\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{2u^{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75e+lvZ+VlbDA71gzzIeWZ0y4+rY5ULRUEksemusM4Yyrrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx4OQzbEnsuafNY3nX9QxDlJhvW8isAw/lkJXdY8g1wM23ql8XXPq6bNQlMm+36iNhkM590FLV/JQkTs6OcJgAWaNH+Jl/Zyy+v2DeiqbV+Z5w==" } }, { "type": "step", "primary": "Substitute back $$u=3x-1$$", "result": "=2\\cdot\\:\\frac{1}{3}\\left(-\\frac{1}{2\\left(3x-1\\right)^{2}}\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{1}{3}\\left(-\\frac{1}{2\\left(3x-1\\right)^{2}}\\right):{\\quad}-\\frac{1}{3\\left(3x-1\\right)^{2}}$$", "input": "2\\cdot\\:\\frac{1}{3}\\left(-\\frac{1}{2\\left(3x-1\\right)^{2}}\\right)", "result": "=-\\frac{1}{3\\left(3x-1\\right)^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-2\\cdot\\:\\frac{1}{3}\\cdot\\:\\frac{1}{2\\left(3x-1\\right)^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=-\\frac{1\\cdot\\:1\\cdot\\:2}{3\\cdot\\:2\\left(3x-1\\right)^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1\\cdot\\:1}{3\\left(3x-1\\right)^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=-\\frac{1}{3\\left(3x-1\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvik6A3oNQUMYwtwrS1tQll7yseZKjdvjH80yyLs20TDfrLTrWWMFI8l4Q07DZ5+hJa8jrIOMU27I/DyX52oGLu0NnfGRHCYkQzO7gHedthJH5as41htHRTP01p4839D55UO5AIz++qluupTlLFEcE9J1NrSQC+Bg7533RohsVU8vlxFTEIqXtitKHBXlxZBmeAkNTCwhUvbgj8dRofLUrgK/sEMVx+fnXXV0/6bikc+l8" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-\\frac{x}{\\left(3x-1\\right)^{2}}-\\left(-\\frac{1}{3\\left(3x-1\\right)^{2}}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-\\frac{x}{\\left(3x-1\\right)^{2}}+\\frac{1}{3\\left(3x-1\\right)^{2}}", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{x}{\\left(3x-1\\right)^{2}}+\\frac{1}{3\\left(3x-1\\right)^{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", "practiceLink": "/practice/integration-practice#area=main&subtopic=Integration%20By%20Parts", "practiceTopic": "Integration by Parts" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{x}{(3x-1)^{2}}+\\frac{1}{3(3x-1)^{2}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }