TITLE: $ \int^{\frac{\pi }{4} }_{0} \cos ^{\frac{3}{2} }\left(2 \theta \right) \cos \left( \theta \right) d\theta $
QUESTION [9 upvotes]: $$ \int^{\frac{\pi }{4} }_{0} \cos ^{\frac{3}{2} }\left(2 \theta \right) \cos \left( \theta \right) d\theta $$
This integration above I tried to solve it by and get
$$ \int^{\frac{\pi }{4} }_{0} \left( 1-2\sin ^{2}\left( \theta \right) \right) ^{\frac{3}{2} }d\sin \left( \theta \right)$$
and I tried to evaluate the power but I find this is useless.
My question is: how I can get this integration in the closed form?
  Thanks.

REPLY [14 votes]: Define $\sin{\phi}=\sqrt{2} \sin{\theta}$.  Then the integral is equal to
$$\frac1{\sqrt{2}} \int_0^{\pi/2} d\phi \, \cos^4{\phi} = \frac{3 \sqrt{2} \pi}{32}$$
NB
$$\int_0^{\pi/2} d\phi \, \cos^{2 n}{\phi} = \frac1{2^{2 n}} \binom{2 n}{n} \frac{\pi}{2} $$