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Question
Integrate the following functions w.r.t. x : sin5x.cos8x
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Solution
Let I = `int sin^5xcos^8xdx`
=`int sin^4xcos^8xsinxdx`
= `int(1 - cos^2x)^2 cos^8xsinxdx`
Put cos x = t
∴ – sin x dx = dt
∴ sin x dx = – dt
I = `- int(1 - t^2)^2t^8 dt`
= `- int(1 - 2t^2 + t^4)t^8 dt`
= `- int (t^8 - 2t^10 + t^12)dt`
= `- int t^8dt + 2 intt^10 dt - int t^12 dt`
= `- t^9/(9) + 2(t^11/11) - t^13/(13) + c`
= `-(1)/(9)cos^9x + (2)/(11)cos^11x - (1)/(13)cos^13x + c`.
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