TITLE: Some clever trick is required for this Integral with irrational power of cosine as integrand.
QUESTION [9 upvotes]: See this:
$$\newcommand{\b}[1]{\left(#1\right)}\left\lfloor\frac{\displaystyle\int_0^{\pi/2}\cos^{\sqrt{13}-1}x{\rm d}x}{\displaystyle\int_0^{\pi/2}\cos^{\sqrt{13}+1}x{\rm d}x}\right\rfloor$$
Well I could only think of Cauchy-Schwarz, but it is also not fitting.
All calculations by hand, no Beta And that implicitly implies not using others like Gamma too.

REPLY [4 votes]: Integration by parts gives, for any $\eta>0$:
$$\int_{0}^{\pi/2}\cos^{\eta+1}(x)\,dx = \frac{\eta}{\eta+1}\int_{0}^{\pi/2}\cos^{\eta-1}(x)\,dx \tag{1}$$
hence by taking $\eta=\sqrt{13}$ we just need to compute:
$$\left\lfloor 1+\frac{1}{\sqrt{13}}\right\rfloor = \color{red}{1}.\tag{2}$$