integral of 1 (x^2)

{ "query": { "display": "$$\\int\\:\\frac{d}{1}\\left(x^{2}\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{d}{1}(x^{2})dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{dx^{3}}{3}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{d}{1}\\left(x^{2}\\right)dx=\\frac{dx^{3}}{3}+C$$", "input": "\\int\\:\\frac{d}{1}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": "=d\\cdot\\:\\int\\:x^{2}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{2}dx", "result": "=d\\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": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{dx^{3}}{3}", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{dx^{3}}{3}+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=Power%20Rule", "practiceTopic": "Integral Power Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{dx^{3}}{3}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }