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?
On
Maple and Mathematica both give answers in terms of elliptic integral functions, which are not elementary functions and so probably don't qualify as "simple closed form". A proof that the antiderivative is not elementary would involve the Risch algorithm, but since this is the mixed transcendental-algebraic case (which isn't even implemented in Maple) I think this would be very complicated.
Depends on what functions you call "standard". It has no elementary antiderivative, but does have one in terms of elliptic functions.