Integral: $\int \sqrt {\sin x} \, \mathrm{d}x$ I want to find $$\int \sqrt {\sin x} \, \mathrm{d}x$$
Now what I think that this can not be integrated without any definite boundary given o.w we can shhift it to gamma function or directly using beta function. But if some one else has any different idea to do this with basic integration technique please share. I also want to post it in integration tag only because I want to see any solution using elementary integration rules(if it exists!!).
 A: It is an elliptic integral (an incomplete elliptic integral of the second kind, exactly) since by replacing $x$ with $\frac{\pi}{2}-2t$ we are left with:
$$ \int \sqrt{\cos(2t)}\,dt = \int\sqrt{1-2\sin^2 t}\,dt.$$
Obviously, there are special values of $E(\varphi,k)$, so the original integral over some intervals leads to nice closed forms, like:
$$ \int_{0}^{\pi/2}\sqrt{\sin x}\,dx = \sqrt{\frac{2}{\pi}}\cdot\Gamma\left(\frac{3}{4}\right)^2.$$
A: Given $\displaystyle \int\sqrt{\sin x}\;dx$
Let $\displaystyle \sin x = t^2\Leftrightarrow \cos xdx = 2tdt\Leftrightarrow dx = \frac{2t}{\sqrt{1-t^4}}dt$
So integral convert into $\displaystyle \int t.\frac{2t}{\left(1-t^4\right)^{\frac{1}{2}}}dt$
So Integral is $\displaystyle 2\int\;t^2.\left(1-t^4\right)^{-\frac{1}{2}}dt$
Now Using $\displaystyle \bullet\; \int x^m.\left(a+bx^n\right)^p\;dx$
where $m\;,n\;,p$ are Rational no. 
which is Integrable only when $\displaystyle \left(\frac{m+1}{n}\right)\in \mathbb{Z}$ or $\displaystyle \left\{\frac{m+1}{n}+p\right\}\in\mathbb{Z}$
Now here $\displaystyle 2\int\;t^2.\left(1-t^4\right)^{-\frac{1}{2}}dt$
$\displaystyle m = 2\;\;,a = 1\;\;,b = -1\;\;,n = 4\;\;,p = -\frac{1}{2}$
and $\displaystyle \left(\frac{2+1}{4}\right)\neq \mathbb{Z}$ or $\displaystyle \left(\frac{2+1}{4}\right)-\frac{1}{2}\neq \mathbb{Z}$
So We can not integrate $\displaystyle \int\sqrt{\sin x}\;dx =2\int\;t^2.\left(1-t^4\right)^{-\frac{1}{2}}dt$ in terms of elementry function.
