Evaluating $\int \sin2x \sqrt{\sin^2x-\cos^3x} \, dx .$

Question

I need some in evaluating the following integral: $$\int \sin2x \sqrt{\sin^2x-\cos^3x} \, dx .$$

Any help would be appreciated.

Answer 1

Write $\sin 2x$ as $2 \sin x \cos x$, which suggests substituting $u = \sin x$ or $u = \cos x$. Before we do this, we must write the expression in the radical just in terms of $\sin$ or $\cos$: The latter is easier, thanks to the Pythagorean identity, $\sin^2 x = 1 - \cos^2 x$, and the resulting form suggests that we choose the substitution $u = \cos x$.

This reduces the integral to $$2 \int u \sqrt{1 - u^2 - u^3} du,$$ and using a CAS shows that the antiderivative involves the (nonelementary) elliptic integral functions (and are a serious mess otherwise too).