integral of 4x-15x^2+e^x

{ "query": { "display": "$$\\int\\:4x-15x^{2}+e^{x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int 4x-15x^{2}+e^{x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "2x^{2}-5x^{3}+e^{x}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:4x-15x^{2}+e^{x}dx=2x^{2}-5x^{3}+e^{x}+C$$", "input": "\\int\\:4x-15x^{2}+e^{x}dx", "steps": [ { "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\\:4xdx-\\int\\:15x^{2}dx+\\int\\:e^{x}dx" }, { "type": "interim", "title": "$$\\int\\:4xdx=2x^{2}$$", "input": "\\int\\:4xdx", "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": "=4\\cdot\\:\\int\\:xdx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=4\\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 $$4\\cdot\\:\\frac{x^{2}}{2}:{\\quad}2x^{2}$$", "input": "4\\cdot\\:\\frac{x^{2}}{2}", "result": "=2x^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x^{2}\\cdot\\:4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV85ueiJEsgPE+iR4wbOw0E77NGoPE9TME3q+OPmgkv2RQ3FzumoH7pkn2g3YS715gwEUqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2SruqOHu1SLz9P47BFOKjaM/6B9qArhIgWv0Fh5tkOCiqL/Q=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:15x^{2}dx=5x^{3}$$", "input": "\\int\\:15x^{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": "=15\\cdot\\:\\int\\:x^{2}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{2}dx", "result": "=15\\cdot\\:\\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" } }, { "type": "interim", "title": "Simplify $$15\\cdot\\:\\frac{x^{3}}{3}:{\\quad}5x^{3}$$", "input": "15\\cdot\\:\\frac{x^{3}}{3}", "result": "=5x^{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x^{3}\\cdot\\:15}{3}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{15}{3}=5$$", "result": "=5x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73QgTbNSVVsaB7tFmYq0peYxRQ4IUY5/J8edDxsXE/Lndd47a0hQ8flDbGsI5To1dbqcMMC9Zn7pVFWRnTSuXqKN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwgA4k4rkwW5mcK+aRy66hSRV4ZAKH9qTsdm9TcPV4HEjI=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "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": "=2x^{2}-5x^{3}+e^{x}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=2x^{2}-5x^{3}+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=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=2x^{2}-5x^{3}+e^{x}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }