TITLE: Evaluating $\int_0^\pi \sqrt{\frac{\sin x}{\pi-x}} \, dx$
QUESTION [8 upvotes]: $$
\int_0^\pi \sqrt{\frac{\sin x}{\pi-x}} \, dx
$$
I have stumbled upon this integral and have no clue how to solve it. 
I know the answer is around $2.2778$. I tried some expansions and some approximations without success. Can you guys figure it out?

REPLY [2 votes]: Can you guys figure it out ?

No. We cannot. And for a very good reason. Basically, you have asked us to evaluate $I\bigg(\dfrac12\bigg),$ 
where $I(a)~=~\displaystyle\int_0^\pi\bigg(\frac{\sin x}x\bigg)^a~dx~=~\frac12~\int_{-\pi}^\pi\bigg(\frac{\sin x}x\bigg)^a~dx.~$ However, even the simpler and 
much more notorious $I(1)~=~\text{Si}(\pi)~$ is not known to possess a meaningful closed form, so it 
would be rather surprising if this one did.