What is the integral `int cos^5 x sin^4 x dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int cos^5x*sin^4x dx` has to be determined.

`int cos^5x*sin^4x dx`

=> `int cos x*cos^4x*sin^4 dx`

=> `int cos x*(1 - sin^2x)*(1 - sin^2x)*sin^4 dx`

let `sin x = y` , `dy = cos x dx`

=> `int(1 - y^2)(1 - y^2)y^4 dy`

=> `int y^4(1 + y^4 - 2y^2) dy`

=> `int y^4 + y^8 - 2*y^6 dy`

=> `y^5/5 + y^9/9 - 2*y^7/7 + C`

Substitute y = sin x

=> `(sin^5x)/5 + (sin^9x)/9 - (2*sin^7x)/7 + C`

The integral `int cos^5x*sin^4x dx` = `(sin^5x)/5 + (sin^9x)/9 - (2*sin^7x)/7 + C`

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nocheooo | Student, Undergraduate | eNotes Newbie

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