What exactly is the average value of the function $f(x)=\cos^{2}x$ on the interval $[0, 2\pi]$? Of course, we assume that the function is integrable on this interval, then we can use the average value theorem $\frac{1}{2\pi}\int_{0}^{2\pi}\cos^2xdx$. I'm not sure about this but is the final value $\frac{3\pi}{4}$? This seems reasonable enough as it is kind of an inflection point.
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$\begingroup$Hints (especially good if you do not have trigonometric identities at hand):
- Note that $\cos^2(x) = \cos(x) * \cos(x)$
- Use integration by parts twice to get an expression of the form $\int_0^{2 \pi} \cos^2(x) \mathrm{d}x = \ldots$ by using $\cos^2(x) = 1-\sin^2(x)$.
- The final value of the average is $1/2$.
Can you take it from here?
Alternatively, you can use the trigonometric identity that Hagen von Eitzen posted in the comments.
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