# Existence of closed form for indefinite integral

During calculus studies, I tried to find a primitive for the following indefinite integral, in a simple form using standard functions:

$\int \sqrt{\sin x} \mathrm{d}x$

I always failed. It may be possible to prove that the primitive exists, or even to find an infinite series expansion for it. But I suspect that it is not possible to find a simple closed-form expression for it.

Am I right? If so, how to prove it?

• I think there is no elementary function whose derivative is your function. Elliptic functions will do it. Jul 19 '13 at 4:35