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Question
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
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Solution
Let I = `int (sinx cos^3x)/(1 + cos^2x).dx`
Put cos x = t
∴ – sin x dx = dt
∴ sin x dx = – dt
I = `- int t^3/(t^2 + 1)dt`
= `- int (t(t^2 + 1) - t)/(t^2 + 1)dt`
= `- int[(t(t^2 + 1))/(t^2 + 1) - t/(t^2 + 1)]dt`
= `- int t dt + int t/(t^2 + 1)dt`
= `- int t dt + (1)/(2) int (2t)/(t^2 + 1)dt`
= `t^2/(2) + (1)/(2)log|t^2 + 1| + c`
... `[∵ d/dt(t^2 + 1) = 2t and int (f'(x))/f(x)dx = log [f(x)] + c]`
= `-(1)/(2) cos^2x + (1)/(2)log|cos^2x + 1| + c`
= `-(1)/(2) cos^2x + (1)/(2)log(1 + cos^2x) + c`
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