I am an Engineer by profession and it has been a long time since I studied Mathematics. I am still confident with my basic skills in Engineering / Applied Maths. Recently, I started tutoring / mentoring a couple of kids in the neighborhood with whatever skills I still possess. They bought me this problem and I am completely stumped.
Any guidance on how to proceed will be greatly appreciated.
$$ Evaluate \left \{ \int_{0}^{\frac{\pi }{2}} \left ( sin\theta \right )^{\frac{3}{4}}d\theta \right \}\cdot\left \{ \int_{0}^{\frac{\pi }{2}} \left ( sin\theta \right )^{-\frac{3}{4}}d\theta \right \} $$
For $a>-1$, $$\int_0^{\pi/2}\sin^a t\,dt=\frac12 B((a+1)/2,1/2) =\frac{\Gamma(a/2+1/2)\Gamma(1/2)}{\Gamma(a/2+1)} =\frac{\sqrt\pi\Gamma((a+1)/2)}{2\Gamma(a/2+1)} $$ where $B$ and $\Gamma$ denote the beta and gamma functions. Your product is $$\frac{\sqrt\pi\Gamma(1/8)}{2\Gamma(5/8)} \frac{\sqrt\pi\Gamma(7/8)}{2\Gamma(11/8)} =\frac{\pi\Gamma(1/8)\Gamma(7/8)}{4(3/8)\Gamma(5/8)\Gamma(3/8)} =\frac{2\pi\sin(3\pi/8)}{3\sin(\pi/8)} =\frac{2\pi}3\tan(3\pi/8) =\frac{2\pi(\sqrt2+1)}{3}.$$