What is : `int cot^5x*csc^3x dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int cot^5x*csc^3x dx` has to be determined.

`int cot^5x*csc^3x dx`

=> `int (cos^5x)/(sin^5x)*(1/sin^3x) dx`

=> `int (cos^5x)/(sin^8x) dx`

=> `int cos x*(1 - sin^2x)^2/(sin^8x) dx`

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

=> `int (1 - y^2)^2/y^8 dy`

=> `int 1/y^8 - 2/y^6 + 1/y^4 dy`

=> `y^-7/-7 + 2*y^-5/5 - y^-3/3`

=> `(-15 + 42y^2 - 35y^4)/(105*y^7)`

substitute y = sin x

=> `(-15 + 42*sin^2x - 35*sin^4x)/(105*sin^7x)`

The integral `int cot^5x*csc^3x dx = (-15 + 42*sin^2x - 35*sin^4x)/(105*sin^7x)`

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