TITLE: Integration of sqrt Sin x dx
QUESTION [12 upvotes]: $$\int \sqrt{\sin x}\ \text dx$$

I asked my teachers and they said that this integration is pretty next level and will be taught in college.
Can anyone help?

REPLY [4 votes]: Let $u=\sqrt{\sin x}$ ,
Then $x=\sin^{-1}u^2$
$dx=\dfrac{2u}{\sqrt{1-u^4}}du$
$\therefore\int\sqrt{\sin x}~dx$
$=\int\dfrac{2u^2}{\sqrt{1-u^4}}du$
$=\int2u^2\sum\limits_{n=0}^\infty\dfrac{(2n)!u^{4n}}{4^n(n!)^2}du$
$=\int\sum\limits_{n=0}^\infty\dfrac{(2n)!u^{4n+2}}{2^{2n-1}(n!)^2}du$
$=\sum\limits_{n=0}^\infty\dfrac{(2n)!u^{4n+3}}{2^{2n-1}(n!)^2(4n+3)}+C$
$=\sum\limits_{n=0}^\infty\dfrac{(2n)!\sin^{2n+\frac{3}{2}}x}{2^{2n-1}(n!)^2(4n+3)}+C$