I stumbled upon this integration $$\int_0^{\pi/4}(1+\cos2\theta)^2\,d\theta.$$ And I have no idea on how to proceed with it. It would be very helpful if someone would provide me with some hints. Thank you.
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2Expanding $(1+\cos(2\theta))^2$ gives $1+2\cos(2\theta) + \cos^2(2\theta)$. The first two terms are straightforward to integrate, for the term $\cos^2(2\theta)$, use the double angle formula! – Chee Han Apr 26 '17 at 05:30
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Using the identity, $\cos^2(2\theta)=\dfrac{1+\cos(4\theta)}{2}$,we can write
$$\begin{align} \int_0^{\pi/4}(1+\cos(2\theta))^2\,d\theta&=\int_0^{\pi/4}(1 + 2 cos(2\theta) + \cos^2(2\theta)) \, d\theta\\[10pt] &=\int_0^{\pi/4} \left(1+2\cos(2\theta)+\frac{1+\cos(4\theta)}{2}\right)\,d\theta \end{align}$$
Can you finish?
Mark Viola
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