integral of ((x-3)^3)/x

{ "query": { "display": "$$\\int\\:\\frac{\\left(x-3\\right)^{3}}{x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{(x-3)^{3}}{x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{x^{3}}{3}-\\frac{9x^{2}}{2}+27x-27\\ln\\left|x\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{\\left(x-3\\right)^{3}}{x}dx=\\frac{x^{3}}{3}-\\frac{9x^{2}}{2}+27x-27\\ln\\left|x\\right|+C$$", "input": "\\int\\:\\frac{\\left(x-3\\right)^{3}}{x}dx", "steps": [ { "type": "interim", "title": "Long division $$\\frac{\\left(x-3\\right)^{3}}{x}:{\\quad}x^{2}-9x+27-\\frac{27}{x}$$", "input": "\\frac{\\left(x-3\\right)^{3}}{x}", "steps": [ { "type": "interim", "title": "Expand $$\\left(x-3\\right)^{3}:{\\quad}x^{3}-9x^{2}+27x-27$$", "input": "\\left(x-3\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply Perfect Cube Formula: $$\\left(a-b\\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}$$", "secondary": [ "$$a=x,\\:\\:b=3$$" ] }, { "type": "step", "result": "=x^{3}-3x^{2}\\cdot\\:3+3x\\cdot\\:3^{2}-3^{3}" }, { "type": "interim", "title": "Simplify $$x^{3}-3x^{2}\\cdot\\:3+3x\\cdot\\:3^{2}-3^{3}:{\\quad}x^{3}-9x^{2}+27x-27$$", "input": "x^{3}-3x^{2}\\cdot\\:3+3x\\cdot\\:3^{2}-3^{3}", "result": "=x^{3}-9x^{2}+27x-27", "steps": [ { "type": "interim", "title": "$$3x^{2}\\cdot\\:3=9x^{2}$$", "input": "3x^{2}\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=9x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7r2B5FIhnmxZhHIA/FXMV1gCWKUbvV6WK3fDUgFtg3Q9sx+gxm3gP8qs3Y7VQpjjAXk2TMpCC2zAuUuL0yAv6CorXsIkMVqPF9sJY22DTip0w+kdEkh7Bcs2dTVHqKRJV" } }, { "type": "interim", "title": "$$3x\\cdot\\:3^{2}=27x$$", "input": "3x\\cdot\\:3^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$3\\cdot\\:3^{2}=\\:3^{1+2}$$" ], "result": "=x\\cdot\\:3^{1+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=27x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AQoFbq5+BRBnIxpwj+cmnACWKUbvV6WK3fDUgFtg3Q/aEIKdHFRryhWgK5J8a/63P8vQyhiD4JSfqjIvcQ7tilO6uzqnvL+LQQSc6cEOzt79Wwv/ClJ19zEnVywdOgYp" } }, { "type": "interim", "title": "$$3^{3}=27$$", "input": "3^{3}", "steps": [ { "type": "step", "primary": "$$3^{3}=27$$", "result": "=27" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Y8AB6W5k6gK6UtOoHj73TQlAlm5MBjmOz6iqN1PySyWF2EOONGHCheUJJ+xtAbTrZYwddaewB9QAIZR+4+aIJCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=x^{3}-9x^{2}+27x-27" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71aw81A/gAOgISE0XQ2VAH913jtrSFDx+UNsawjlOjV3PZoeVRFoTqEIL2MdZ6ZubaVzSFKHwgQq7LcPkCJKHt4EFMST8lDZxn1Yq5HMKVTvhcTtYobM4LbUNcYCRiXOPRLjYUVGnPJmEtxdqz+u5Cg==" } }, { "type": "step", "result": "=\\frac{x^{3}-9x^{2}+27x-27}{x}" }, { "type": "interim", "title": "Divide $$\\frac{x^{3}-9x^{2}+27x-27}{x}:{\\quad}\\frac{x^{3}-9x^{2}+27x-27}{x}=x^{2}+\\frac{-9x^{2}+27x-27}{x}$$", "result": "=x^{2}+\\frac{-9x^{2}+27x-27}{x}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$x^{3}-9x^{2}+27x-27$$<br/>and the divisor $$x\\::\\:\\frac{x^{3}}{x}=x^{2}$$", "result": "\\mathrm{Quotient}=x^{2}" }, { "type": "step", "primary": "Multiply $$x$$ by $$x^{2}:\\:x^{3}$$", "secondary": [ "Subtract $$x^{3}$$ from $$x^{3}-9x^{2}+27x-27$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-9x^{2}+27x-27" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x^{3}-9x^{2}+27x-27}{x}=x^{2}+\\frac{-9x^{2}+27x-27}{x}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-9x^{2}+27x-27}{x}:{\\quad}\\frac{-9x^{2}+27x-27}{x}=-9x+\\frac{27x-27}{x}$$", "result": "=x^{2}-9x+\\frac{27x-27}{x}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-9x^{2}+27x-27$$<br/>and the divisor $$x\\::\\:\\frac{-9x^{2}}{x}=-9x$$", "result": "\\mathrm{Quotient}=-9x" }, { "type": "step", "primary": "Multiply $$x$$ by $$-9x:\\:-9x^{2}$$", "secondary": [ "Subtract $$-9x^{2}$$ from $$-9x^{2}+27x-27$$ to get new remainder" ], "result": "\\mathrm{Remainder}=27x-27" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-9x^{2}+27x-27}{x}=-9x+\\frac{27x-27}{x}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{27x-27}{x}:{\\quad}\\frac{27x-27}{x}=27+\\frac{-27}{x}$$", "result": "=x^{2}-9x+27+\\frac{-27}{x}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$27x-27$$<br/>and the divisor $$x\\::\\:\\frac{27x}{x}=27$$", "result": "\\mathrm{Quotient}=27" }, { "type": "step", "primary": "Multiply $$x$$ by $$27:\\:27x$$", "secondary": [ "Subtract $$27x$$ from $$27x-27$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-27" }, { "type": "step", "primary": "Therefore", "result": "\\frac{27x-27}{x}=27+\\frac{-27}{x}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "step", "primary": "Simplify", "result": "=x^{2}-9x+27-\\frac{27}{x}" } ], "meta": { "solvingClass": "Long Division", "interimType": "Algebraic Manipulation Long Division Title 1Eq" } }, { "type": "step", "result": "=\\int\\:x^{2}-9x+27-\\frac{27}{x}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": "=\\int\\:x^{2}dx-\\int\\:9xdx+\\int\\:27dx-\\int\\:\\frac{27}{x}dx" }, { "type": "interim", "title": "$$\\int\\:x^{2}dx=\\frac{x^{3}}{3}$$", "input": "\\int\\:x^{2}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{2}dx", "result": "=\\frac{x^{3}}{3}", "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^{2+1}}{2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{2+1}}{2+1}:{\\quad}\\frac{x^{3}}{3}$$", "input": "\\frac{x^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{x^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{3}}{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+w+ikB2VyJnNfLrQuoxvVyo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7odVISTIak7VD9OG2tlObqsigQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:9xdx=\\frac{9x^{2}}{2}$$", "input": "\\int\\:9xdx", "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": "=9\\cdot\\:\\int\\:xdx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=9\\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": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{9x^{2}}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:27dx=27x$$", "input": "\\int\\:27dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=27x" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{27}{x}dx=27\\ln\\left|x\\right|$$", "input": "\\int\\:\\frac{27}{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": "=27\\cdot\\:\\int\\:\\frac{1}{x}dx" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{x}dx=\\ln\\left(\\left|x\\right|\\right)$$", "result": "=27\\ln\\left|x\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{x^{3}}{3}-\\frac{9x^{2}}{2}+27x-27\\ln\\left|x\\right|" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{x^{3}}{3}-\\frac{9x^{2}}{2}+27x-27\\ln\\left|x\\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=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{x^{3}}{3}-\\frac{9x^{2}}{2}+27x-27\\ln\\left|x\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }