TITLE: Evaluate $ \int_{0}^{\pi/4}\left(\cos 2x \right)^{11/2}\cdot \cos x\;dx $
QUESTION [11 upvotes]: Evaluate the definite integral
$$
I=\int_{0}^{\pi/4}(\cos 2x)^{11/2}\cdot \cos x\;dx
$$

My Attempt:
$$
I = \int \left(1-2\sin^2 x\right)^{11/2}\cdot \cos x\;dx
$$
Now, substitute $\sin x=t$ with $\cos x \,dx = dt$:
$$
I = \int (1-2t^2)^{11/2}\;dt
$$
How can I complete the solution from this point?

REPLY [3 votes]: Indeed, this special functions approach may be generalized.
Set $$j(s)=\int_0^{\pi/4}\cos(2x)^s\cos(x)dx.$$
Use $t=2\sin(x)^2$ so that $dt=4\sin(x)\cos(x)dx$, i.e. $\cos(x)dx=(2t)^{-1/2}dt$. Hence we have
$$j(s)=\frac1{\sqrt2}\int_0^1t^{1/2-1}(1-t)^{s+1-1}dt=\sqrt{\frac\pi2}\frac{\Gamma(s+1)}{\Gamma(s+\frac32)},$$
as $$\int_0^1t^{a-1}(1-t)^{b-1}dt=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$
and $\Gamma(\tfrac12)=\sqrt\pi$.