Find the integral `int sec^4 x*tan^5 x dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int sec^4 x*tan^5 x dx` has to be determined

 `int sec^4 x*tan^5 x dx`

=> `int (1/(cos^4x))*(sin^5x)/(cos^5x) dx`

=> `int (sin^5x)/(cos^9x) dx`

=> `int sin x*((1 - cos^2x)^2)/(cos^9x) dx`

let `cos x = y => dy = -sin x dx`

=> `-int ((1 - y^2)^2)/y^9 dy`

=> `-int (1 + y^4 - 2y^2)/y^9 dy`

=> `-int 1/y^9 +1/y^5 - 2/y^7 dy`

=> `y^-8/8 + y^-4/4 - y^-6/3`

substitute y = cos x

=> `1/(8*cos^8x) + 1/(4*cos^4x) - 1/(3*cos^6x) + C`

The integral `int sec^4 x*tan^5 x dx = 1/(8*cos^8x) + 1/(4*cos^4x) - 1/(3*cos^6x) + C`

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