integral of 4sin(x)cos^2(x)

{ "query": { "display": "$$\\int\\:4\\sin\\left(x\\right)\\cos^{2}\\left(x\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int 4\\sin(x)\\cos^{2}(x)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{4}{3}\\cos^{3}(x)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:4\\sin\\left(x\\right)\\cos^{2}\\left(x\\right)dx=-\\frac{4}{3}\\cos^{3}\\left(x\\right)+C$$", "input": "\\int\\:4\\sin\\left(x\\right)\\cos^{2}\\left(x\\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": "=4\\cdot\\:\\int\\:\\sin\\left(x\\right)\\cos^{2}\\left(x\\right)dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\sin\\left(x\\right)\\cos^{2}\\left(x\\right)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: $$u=\\cos\\left(x\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-\\sin\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$", "result": "=-\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoTIPsH/5VFEfonU6bvi80j8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zA92cpyjnQxeYFWLLJRXAqw02ZR5clxTmOwI/5g0CzzvDtz8RMf2ztf85Qhda6goD78yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-\\sin\\left(x\\right)dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\sin\\left(x\\right)u^{2}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)du" }, { "type": "interim", "title": "Simplify $$\\sin\\left(x\\right)u^{2}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right):{\\quad}-u^{2}$$", "input": "\\sin\\left(x\\right)u^{2}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\sin\\left(x\\right)u^{2}\\frac{1}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:\\sin\\left(x\\right)u^{2}}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sin\\left(x\\right)$$", "result": "=-1\\cdot\\:u^{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:u^{2}=u^{2}$$", "result": "=-u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-u^{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7yjWNUABL1ljUERrDEpgq1Z/Y6Ysu7WowP0srzOk/hMGcSPKmf0CNKWC8kGIkCk3X3KtxwM1n7owdoG2GFz6ksdsfL2GYVRvzzfVlswjxO2Y2i4NSfL92YiZ5FOsSEDy8nql8XXPq6bNQlMm+36iNhkkjuzIgeJUg10ybKgq0r22txEId7lZcSHdTAsAvmTZFg==" } }, { "type": "step", "result": "=4\\cdot\\:\\int\\:-u^{2}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": "=4\\left(-\\int\\:u^{2}du\\right)" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:u^{2}du", "result": "=4\\left(-\\frac{u^{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{2+1}}{2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{2+1}}{2+1}:{\\quad}\\frac{u^{3}}{3}$$", "input": "\\frac{u^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{u^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{u^{3}}{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7ztMnkJqJUs4Qp2osi+oqZ+o/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ofEE1c1KRzl1GtEOEJ8QtOggQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "=4\\left(-\\frac{\\cos^{3}\\left(x\\right)}{3}\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(-\\frac{\\cos^{3}\\left(x\\right)}{3}\\right):{\\quad}-\\frac{4}{3}\\cos^{3}\\left(x\\right)$$", "input": "4\\left(-\\frac{\\cos^{3}\\left(x\\right)}{3}\\right)", "result": "=-\\frac{4}{3}\\cos^{3}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-4\\cdot\\:\\frac{\\cos^{3}\\left(x\\right)}{3}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{\\cos^{3}\\left(x\\right)\\cdot\\:4}{3}" }, { "type": "step", "result": "=-\\frac{4}{3}\\cos^{3}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rWBXRCto8fcsWph4SfT1wb/2fSuXXEf1orVuUoi5cJnehkKrn0era9rz8TlL+x/vttdvQxZI3PlVepHWO3+UgkgIWpnifm5ipINTTvdEcVA45vezAXqyiEBFLUQXYhP47kAjP76qW66lOUsURwT0neSKCH6iwkTLw2NzbuUZwiseZKmDIqdbj+5O2fTCBD2V4gBJl4WMO1rA0a30/bUYlg==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{4}{3}\\cos^{3}\\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=Trig%20Power%20Multiplication", "practiceTopic": "Integral Trig Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{4}{3}\\cos^{3}(x)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }