integral of-cos^3(x)

{ "query": { "display": "$$\\int\\:-\\cos^{3}\\left(x\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int -\\cos^{3}(x)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\sin(x)+\\frac{1}{3}\\sin^{3}(x)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:-\\cos^{3}\\left(x\\right)dx=-\\sin\\left(x\\right)+\\frac{1}{3}\\sin^{3}\\left(x\\right)+C$$", "input": "\\int\\:-\\cos^{3}\\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": "=-\\int\\:\\cos^{3}\\left(x\\right)dx" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\int\\:\\cos^{3}\\left(x\\right)dx", "result": "=-\\int\\:\\left(1-\\sin^{2}\\left(x\\right)\\right)\\cos\\left(x\\right)dx", "steps": [ { "type": "interim", "title": "Simplify $$\\cos^{3}\\left(x\\right):{\\quad}\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)$$", "input": "\\cos^{3}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "secondary": [ "$$\\cos^{3}\\left(x\\right)=\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)$$" ], "result": "=\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dmwH+NFlXzYMc8KrQqkQP1XTSum/z5kLpMzXS1UJIey3kQDEAk6UuWKo1vzeBjzjNgPgSkZkNSKp8+YLNOQRCGRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9vIRDjLY3QM6NIhDAK43m0SS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\int\\:\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)dx" }, { "type": "step", "primary": "Use the following identity: $$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$", "result": "=\\int\\:\\left(1-\\sin^{2}\\left(x\\right)\\right)\\cos\\left(x\\right)dx" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xOu/kODwo/3r139yZjXeWP1fWutlCU4zUzgkFchUVlKOeWgsE4Mk40prEDZkKQ8Xcq64+b8YguXf4qCtKW9b7JFhxSzNcjgPzX10I7nnzbP34i0lW0DrXT5HrSgF4N/aPsSAighYt+7xJRvPKyzPBBFKk3fejFkyiOiq9iG9IkAO/B7UtNBfw19bDRpX2AlzDEZOhzOLFFCWOVe0TAs9Ps/tVtg7k21KzJIFYRgfW7H" } }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\left(1-\\sin^{2}\\left(x\\right)\\right)\\cos\\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: $$u=\\sin\\left(x\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\cos\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOt2FhQQwx0GxLGzv2mPOv8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaugB66mSUqneplfTkjggryzD6iDpcTVxDjQ5tzND5SL/a/YPuLKiL1T3raVYWYuVCmA==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\cos\\left(x\\right)dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{\\cos\\left(x\\right)}du$$" }, { "type": "step", "result": "=\\int\\:\\left(1-u^{2}\\right)\\cos\\left(x\\right)\\frac{1}{\\cos\\left(x\\right)}du" }, { "type": "interim", "title": "Simplify $$\\left(1-u^{2}\\right)\\cos\\left(x\\right)\\frac{1}{\\cos\\left(x\\right)}:{\\quad}1-u^{2}$$", "input": "\\left(1-u^{2}\\right)\\cos\\left(x\\right)\\frac{1}{\\cos\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\left(1-u^{2}\\right)\\cos\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\cos\\left(x\\right)$$", "result": "=1\\cdot\\:\\left(1-u^{2}\\right)" }, { "type": "step", "primary": "Refine", "result": "=1-u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:1-u^{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+mA3m8h0M8PsnqGscp8ZqUct8+f6Y9+RLVqXXO+g6VKB42dCZngAt2UhJjMVkTWLsN5vRWJvr0V/8T6RtR0CSAgHP4YqWkh5OOHaSrQ/l8R8tW3miTn3iPAMl3Bw27YmUUqTd96MWTKI6Kr2Ib0iQBZegS2gwh8pq/gwNfwaDRbAwNT33I9ftOGdlyhcHnXlA==" } }, { "type": "step", "result": "=-\\int\\:1-u^{2}du" }, { "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": "=-\\left(\\int\\:1du-\\int\\:u^{2}du\\right)" }, { "type": "interim", "title": "$$\\int\\:1du=u$$", "input": "\\int\\:1du", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:u" }, { "type": "step", "primary": "Simplify", "result": "=u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:u^{2}du=\\frac{u^{3}}{3}$$", "input": "\\int\\:u^{2}du", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:u^{2}du", "result": "=\\frac{u^{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{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" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-\\left(u-\\frac{u^{3}}{3}\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "=-\\left(\\sin\\left(x\\right)-\\frac{\\sin^{3}\\left(x\\right)}{3}\\right)" }, { "type": "interim", "title": "Simplify $$-\\left(\\sin\\left(x\\right)-\\frac{\\sin^{3}\\left(x\\right)}{3}\\right):{\\quad}-\\sin\\left(x\\right)+\\frac{1}{3}\\sin^{3}\\left(x\\right)$$", "input": "-\\left(\\sin\\left(x\\right)-\\frac{\\sin^{3}\\left(x\\right)}{3}\\right)", "result": "=-\\sin\\left(x\\right)+\\frac{1}{3}\\sin^{3}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(\\sin\\left(x\\right)\\right)-\\left(-\\frac{\\sin^{3}\\left(x\\right)}{3}\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-\\sin\\left(x\\right)+\\frac{\\sin^{3}\\left(x\\right)}{3}" }, { "type": "step", "result": "=-\\sin\\left(x\\right)+\\frac{1}{3}\\sin^{3}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Mdrn+5z6dmEt7LdSLpxXY1wCt/NJvDQc1iUcHQYK9rOpKaNLc+hAe/2EsvyRP8CAcJChiVhDxT5N/LHSTLMjyHgV5farxNddjHBnDxPCCz1b0VkN4Ux/QR5FDN/rpvIqtI3McHOf67QvdoFo+ZeWcpjcBIL5pmo83UMFZRSzJsVck/b4Gmnsj+yIWJXYK8ckuNZ6n6URJRCC6Mjg5vKY5e2oSFq7CefveIY0INpkJ3I=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\sin\\left(x\\right)+\\frac{1}{3}\\sin^{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=-\\sin(x)+\\frac{1}{3}\\sin^{3}(x)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }