Question

Integrate the following functions w.r.t.x:

cos⁸xcotx

Answer 1

Let I = `int cos^8xcotxdx`

= `int cos^8x. cosx/sinx .dx`

Put sinx = t

∴ cosx dx = dt

cos⁸x = (cos²x)⁴

= (1 – sin²x)⁴

= (1 – t²)⁴

= 1 –  4t² + 6t⁴ – 4t⁶ + t⁸

I = `int(1 - 4t^2 + 6t^4 - 4t^6 + t^8)/tdt`

= `int[1/t - 4t +6t^3 - 4t^5 + t^7]dt`

= `int 1/t dx - 4 int tdt + 6 int t^3 dt - 4 int t^5 dt + int t^7 dt`

= `log|t| - 4 (t^2/2) + 6(t^4/4) - 4(t^6/6) + t^8/(8) + c`

= `log|sinx| - 2sin^2x + 3/2 sin^4x - 2/3 sin^6x + (sin^8x)/(8) + c`.