How to find $ \int\sqrt{\sin x\cos x}\,dx $

Question

I have been working on days to find the integral of the following question: $$ \int\sqrt{\sin x\cos x}\,dx $$ Any anyone please help in finding the solution of that question?

Answer 1

I have been working on days to find the integral of the following question

Even if you were to have kept on working on it for years, you still would not have been able to reach any conclusion, since it cannot be expressed in terms of elementary functions. See Liouville's theorem and the Risch algorithm for more information. However, we have $$\int_0^\tfrac\pi4\sqrt{\sin x\cos x}~dx~=~\int_\tfrac\pi4^\tfrac\pi2\sqrt{\sin x\cos x}~dx~=~\frac12\int_0^\tfrac\pi2\sqrt{\sin x\cos x}~dx~=~\dfrac{\pi\sqrt\pi}{\Gamma^2\bigg(\dfrac14\bigg)}$$ See Wallis' integrals, and their relation to the beta and $\Gamma$ function for more details.

Answer 2

Wolfram gives the following result :

http://www.wolframalpha.com/input/?i=integral+sqrt%28sin%28x%29cos%28x%29%29

As you can see, there is no "nice" antiderivate to $\sqrt{sin(x)cos(x)}$