It seems like I am stuck on such a simple problem:
How to I find the antiderivative of $\cos^2x$? I have tried partial integration, it doesn't seem to work (for me). Some help on how to integrate it would be nice.
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4Use $\cos2x=2\cos^2x-1$. – almagest Jun 15 '16 at 13:16
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3Similar older question: http://math.stackexchange.com/questions/1628840/how-do-i-solve-int4-cos2x-dx – Martin Sleziak Jun 17 '16 at 04:19
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3And one explaining how to do ALL integrals $\int\cos^mx\sin^nx,dx$. – Jyrki Lahtonen Jun 17 '16 at 04:52
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I don't know the exact policy on when to mark a question as duplicate, but I wouldn't refer someone with this particular question to Evaluating $\int P(\sin x, \cos x) \text{d}x$. Having to go through the general Weierstrass substitution or reduction formulas is far too complicated for a basic integral like this; the standard approach for this one isn't even mentioned in that general answer. I agree that the link Martin Sleziak provides is indeed very similar although the OP here also tried integration by parts and got a specific answer for it. – StackTD Jun 20 '16 at 12:13
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Perhaps it is worth mentioning that this specific question was mentioned as an example in this discussion on meta about duplicates. – Martin Sleziak Mar 22 '17 at 19:12
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See also: How to evaluate $\int \cos^2x \ dx$ – Martin Sleziak Jul 09 '17 at 08:53
2 Answers
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You use the identity (e.g. solving from $\cos(2x) = 2\cos^2x-1$): $$\cos^2 x = \frac{1+ \cos(2x)}{2}$$
Addendum: the previous hint will give you the easiest solution, but you mentioned an attempt with integration by parts - that would work too: $$\begin{array}{rl} \displaystyle \color{blue}{\int \cos^2x \, \mbox{d}x} & \displaystyle = \int \cos x \, \mbox{d} \sin x \\[6pt] & \displaystyle = \cos x \sin x - \int \sin x \, \mbox{d} \cos x \\[6pt] & \displaystyle = \cos x \sin x + \int \sin^2 x \, \mbox{d}x \\[6pt] & \displaystyle = \cos x \sin x + \int 1-\cos^2 x \, \mbox{d}x \\[6pt] & \displaystyle = \cos x \sin x + x - \color{blue}{\int \cos^2x \, \mbox{d}x} \\[6pt] \Rightarrow \displaystyle 2 \int \cos^2x \, \mbox{d}x & \displaystyle = \cos x \sin x + x + C \end{array}$$ Then you divide by 2.
StackTD
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$$\int (\cos x)^2\, dx=\int \frac{1+\cos(2x)}{2}\, dx$$
$$=\frac{1}{2}\left(\int \, dx+\frac{1}{2}\int \cos(2x)\, d(2x)\right)$$
$$=\frac{1}{2}\left(x+\frac{1}{2}\sin(2x)\right)+C=\frac{x}{2}+\frac{\sin(2x)}{4}+C$$
user236182
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