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$\ I_n = \int \cos^n \theta d\theta$

Find a recursive equation to express $\ I_n $ in terms of $\ I_n-_2 $. Find $\ I_6 $ and $\ I_7 $ as well.

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en.wikipedia.org/wiki/… –  TenaliRaman Nov 23 '12 at 17:34
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2 Answers

Hint: $\cos^n(x) = \cos(x)\cos^{n-1}(x)$ and try integrating by parts. Also: remember that $\sin^2(x) + \cos^2(x) = 1$

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We show how to avoid explicit integration by parts. Guess that $f(x)=\cos^{n-1} x\sin x$ works. Check by differentiating. We get $$f'(x)= \cos^n x-(n-1)\sin^2 x\,\cos^{n-2} x.$$ The first part looks good. For the second part, write $\sin^2 x=1-\cos^2 x$.

We get $$f'(x)=n\cos^n x-(n-1)\cos^{n-2} x.$$ It follows that $$n\int \cos^n x\,dx=\cos^{n-1}x\, \sin x +(n-1)\int \cos^{n-2} x\,dx.$$

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