integral of cos^2(x)tan^3(x)

{ "query": { "display": "$$\\int\\:\\cos^{2}\\left(x\\right)\\tan^{3}\\left(x\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\cos^{2}(x)\\tan^{3}(x)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\ln\\left|\\cos(x)\\right|+\\frac{\\cos^{2}(x)}{2}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\cos^{2}\\left(x\\right)\\tan^{3}\\left(x\\right)dx=-\\ln\\left|\\cos\\left(x\\right)\\right|+\\frac{\\cos^{2}\\left(x\\right)}{2}+C$$", "input": "\\int\\:\\cos^{2}\\left(x\\right)\\tan^{3}\\left(x\\right)dx", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\int\\:\\cos^{2}\\left(x\\right)\\tan^{3}\\left(x\\right)dx", "result": "=\\int\\:\\frac{\\left(1-\\cos^{2}\\left(x\\right)\\right)\\sin\\left(x\\right)}{\\cos\\left(x\\right)}dx", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\tan\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=\\int\\:\\cos^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{3}dx" }, { "type": "interim", "title": "Simplify $$\\cos^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{3}:{\\quad}\\frac{\\sin^{3}\\left(x\\right)}{\\cos\\left(x\\right)}$$", "input": "\\cos^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{3}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{3}=\\frac{\\sin^{3}\\left(x\\right)}{\\cos^{3}\\left(x\\right)}$$", "input": "\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\sin^{3}\\left(x\\right)}{\\cos^{3}\\left(x\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73yQ3tP4bkGLx3qZ5P//Et+wNnQzv5E758+LKB991QDbk9CU8MCWpvAxSxvbI1SlD/aL2Coo0GISQwm8bv5wDiS3jrVB3uPQYc1Y44VXRtP/FxjSg0Dgjtlmj67WMMkYz/z//r+dXk7h9vxeDCLuZqn2Nii2xSdcID3T1P97D5/LWbjv5iVhsnEuGNCIzdVjSxPJzd9XLwQ6NF+TMQiN1CEgHhc2mpuBTGSYqT859RzTbhdvJ0gUtL6/ZANEjcKM+" } }, { "type": "step", "result": "=\\frac{\\sin^{3}\\left(x\\right)}{\\cos^{3}\\left(x\\right)}\\cos^{2}\\left(x\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sin^{3}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{3}\\left(x\\right)}" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{\\cos^{2}\\left(x\\right)}{\\cos^{3}\\left(x\\right)}=\\frac{1}{\\cos^{3-2}\\left(x\\right)}$$" ], "result": "=\\frac{\\sin^{3}\\left(x\\right)}{\\cos^{3-2}\\left(x\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$3-2=1$$", "result": "=\\frac{\\sin^{3}\\left(x\\right)}{\\cos\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{\\sin^{3}\\left(x\\right)}{\\cos\\left(x\\right)}dx" }, { "type": "interim", "title": "Simplify $$\\sin^{3}\\left(x\\right):{\\quad}\\sin^{2}\\left(x\\right)\\sin\\left(x\\right)$$", "input": "\\sin^{3}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "secondary": [ "$$\\sin^{3}\\left(x\\right)=\\sin^{2}\\left(x\\right)\\sin\\left(x\\right)$$" ], "result": "=\\sin^{2}\\left(x\\right)\\sin\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yfOMpqtoyVbyzBuY4dZ1clXTSum/z5kLpMzXS1UJIewjFOcsU8TXhzIvoMfbQqJyy639uHgZUk3OvF5lwCd8YWRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9OTqelqHtNcOqee5KfeSa2CS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\int\\:\\frac{\\sin^{2}\\left(x\\right)\\sin\\left(x\\right)}{\\cos\\left(x\\right)}dx" }, { "type": "step", "primary": "Use the following identity: $$\\sin^{2}\\left(x\\right)=1-\\cos^{2}\\left(x\\right)$$", "result": "=\\int\\:\\frac{\\left(1-\\cos^{2}\\left(x\\right)\\right)\\sin\\left(x\\right)}{\\cos\\left(x\\right)}dx" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77vvHnQZ9ZHvmNRHJX0hVA7pvwFSE/LcWCwl2+R5KG/L0ZWU94MxLy/oE35gBI0JRW+LsMhiNpJoBfxygMW1qiGLbYolnlfVBg1rM7gSmSGCrNA6f/Whry8gYdzHmfqAbGKdD3MLOx9K4xIiGPoI0kZlyNjnPmegSXEWznOXFhW1qFqxYOtV4/zU/W5FpXnjToEFMST8lDZxn1Yq5HMKVTsNwWCajYaZ4bYqH0RpI9A/zFCnzKTYR2+JO9+HPqu4771FBr5UXMxw7y43bLdm5aQ=" } }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\left(1-\\cos^{2}\\left(x\\right)\\right)\\sin\\left(x\\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: $$u=\\cos\\left(x\\right)$$" ] }, { "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\\:\\frac{\\left(1-u^{2}\\right)\\sin\\left(x\\right)}{u}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{\\left(1-u^{2}\\right)\\sin\\left(x\\right)}{u}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right):{\\quad}-\\frac{1-u^{2}}{u}$$", "input": "\\frac{\\left(1-u^{2}\\right)\\sin\\left(x\\right)}{u}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{\\left(1-u^{2}\\right)\\sin\\left(x\\right)}{u}\\cdot\\:\\frac{1}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{\\left(1-u^{2}\\right)\\sin\\left(x\\right)\\cdot\\:1}{u\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sin\\left(x\\right)$$", "result": "=-\\frac{\\left(1-u^{2}\\right)\\cdot\\:1}{u}" }, { "type": "interim", "title": "$$\\left(1-u^{2}\\right)\\cdot\\:1=1-u^{2}$$", "input": "\\left(1-u^{2}\\right)\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply: $$\\left(1-u^{2}\\right)\\cdot\\:1=\\left(1-u^{2}\\right)$$", "result": "=\\left(1-u^{2}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=1-u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xUDavLaCgRpkqVkgP5YRNWiBFxjfQHIgU3vZ8lSvjvOjkVi15I8rBefLi4Iyt2wrbOGdL5kbx8TrQ9KiufT7dw/+VDr0iUOQpsxwi5T5Q7SRjFma7+UFIWeHulULG5TjsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=-\\frac{-u^{2}+1}{u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-\\frac{1-u^{2}}{u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s72KdD3MLOx9K4xIiGPoI0kZlyNjnPmegSXEWznOXFhW10nmHggi5+2YhiZrFyYMJIyue+lhDRrKAl/Ka6Jx2qlkpUI/L4vIdDW82QAXbXJFbsZ7y+39sI2nPvCj6MIgLyqU+GfAEtCVqXqSsgOO13YtFQkMLhO2dLK/62ZNxQ5ThgQUxJPyUNnGfVirkcwpVOwYt4zFWXnnExjER+MoFq26VxBSUw6GSpC8X07xDz9nm" } }, { "type": "step", "result": "=\\int\\:-\\frac{1-u^{2}}{u}du" }, { "type": "interim", "title": "Expand $$-\\frac{1-u^{2}}{u}:{\\quad}-\\frac{1}{u}+u$$", "input": "-\\frac{1-u^{2}}{u}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{1-u^{2}}{u}=-\\left(\\frac{1}{u}\\right)-\\left(-\\frac{u^{2}}{u}\\right)$$" ], "result": "=-\\left(\\frac{1}{u}\\right)-\\left(-\\frac{u^{2}}{u}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a,\\:-\\left(-a\\right)=a$$", "result": "=-\\frac{1}{u}+\\frac{u^{2}}{u}" }, { "type": "interim", "title": "Cancel $$\\frac{u^{2}}{u}:{\\quad}u$$", "input": "\\frac{u^{2}}{u}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=u" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpOpJcQApkTZ+uMFk1dFeJTNGoPE9TME3q+OPmgkv2RQcjH6Om9I8v7pj2lJ7T2vN5zRQCb3t7dl0WzvSMzsBj7zXB/plvJ2hdDMbGWW2EJV5+CYbobceaqVsQD6172CVQ==" } }, { "type": "step", "result": "=-\\frac{1}{u}+u" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g4dlyxAZpJgxorhm9VHUU/tN3a1x7wFP+Pe2a2MwAL5wkKGJWEPFPk38sdJMsyPIN286EGQxTwZxO92BEjgNL1o7Pn5aD3wa1H1ZoSiJ48f8bYA0b6V2RSTOZ7Os9NOD3cmk0SR4IB/31/W7z8ixv2RpspkIXbEaDUX84Ey2ZCE=" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{u}+udu" }, { "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\\:\\frac{1}{u}du+\\int\\:udu" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{u}du=\\ln\\left|u\\right|$$", "input": "\\int\\:\\frac{1}{u}du", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=\\ln\\left|u\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xYpRYRjORiCJkHaCCs/U61fRzuAubUUJYHDgzAJMdz/8+LBiPyAP34u+4MuPtfQQvGeKNchbzBJJgE/Z7UYTEUZH/yLEwzXgrACX51mM8dC3KorEA/Q0CbMb/fBf5r55LRNvOTr4fZAVVYGYBAkbV86k5tT+RaEjrsoFUO/AsBKtTI+06tguYsUGJG/KKWkTQ==" } }, { "type": "interim", "title": "$$\\int\\:udu=\\frac{u^{2}}{2}$$", "input": "\\int\\:udu", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:udu", "result": "=\\frac{u^{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{u^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{1+1}}{1+1}:{\\quad}\\frac{u^{2}}{2}$$", "input": "\\frac{u^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77irOeniMfrJKKN+TrhAcvL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpAUgTzPSdH5PWV4NCtvwjA7/YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-\\ln\\left|u\\right|+\\frac{u^{2}}{2}" }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "=-\\ln\\left|\\cos\\left(x\\right)\\right|+\\frac{\\cos^{2}\\left(x\\right)}{2}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\ln\\left|\\cos\\left(x\\right)\\right|+\\frac{\\cos^{2}\\left(x\\right)}{2}+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=-\\ln\\left|\\cos(x)\\right|+\\frac{\\cos^{2}(x)}{2}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }