Q:

Integrate $\int \frac{\sqrt x}{\sqrt {a^2-x^2}}dx$

I need to evaluate the integral $$\int \frac{\sqrt x}{\sqrt {a^2-x^2}}dx$$
I substituted $x=a\sin\theta$
Hence, the required integral is reduced to 
$$\sqrt a \int \sqrt {\sin\theta}d\theta$$
However the integration of this function yields an elliptic function. Is there any way to integrate it so that it gives a more elementary function?

A:

The antiderivative of a function is unique up to a constant. So, no, if you have found that the integral evaluates to an elliptic function, it will always evaluate to that elliptic function plus a constant no matter what method you use.

