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Question
Integrate the following functions w.r.t. x : cos7x
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Solution
Let I = `int cos^7x dx`
= `int cos^6x.cos x dx`
= `int (1 - sin^2x)^3 cos x dx`
Put, sin x = t
∴ cos x dx = dt
I = `int(1 - t^2)^3 dt`
= `int(1 - 3t^2 + 3t^4 - t^6)dt`
= `int 1dt - 3 int t^2dt + 3intt^4 dt - int t^6 dt`
= `t - 3(t^3/3) + 3(t^5/3) - t^7/(7) + c`
= `sinx - sin^3x + (3)/(5)sin^5x - (1)/(7)sin^7x + c`.
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