# Integrate $\int \frac{\sqrt x}{\sqrt {a^2-x^2}}dx$ [duplicate]

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?

• No---see math.stackexchange.com/questions/1469846/… (of which this question is essentially a duplicate). More generally: Substituting gives the equivalent integral $$\int \frac{u^2 \,du}{\sqrt{a^2 - u^4}} ,$$ for but integrals of the form $$\int \frac{p(u) \,du}{\sqrt{q(u)}}$$ with $p, q$ polynomials and $q$ cubic or quartic, expressing the antiderivative requires elliptic functions or the equivalent to express, except in special cases that $p, q$ are related in certain ways. Commented Sep 10, 2019 at 17:11

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...