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

Question

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?

Answer 1

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.

Answer 2

if we find an analytic way to integrate what you are asking, wouldn't that automatically become a good way to integrate $\sqrt{\sin \theta}$? i don't think there is a good way to integrate elliptic functions...