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Question
Integrate the following functions w.r.t.x:
cos8xcotx
Solution
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`.
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