Stuck on the integral: $\int x^2(1-x^2)^{1/2}\, dx$

Question

So far I've substituted $x=\sin t$ ; $dx = \cos t\; dt$, leaving me to integrate $\sin^2t\cos^2t\;dt$.

I'm stuck here. I thought to use the identity $\sin 2t = 2 \sin t \cos t$ but it looks like it doesn't lead anywhere.

Any tips would be greatly appreciated!

Answer 1

Note that $\sin(2t) = 2\sin(t)\cos(t)$ and $\cos(2t) = 1-2\sin^2(t)$. It follows that

$$\int x^2\sqrt{1-x^2}dx = \int \sin^2(t)\cos^2(t) dt = \int (\frac{1}{2}\sin(2t))^2dt =\frac{1}{4} \int \frac{1-\cos(4t)}{2}dt\text{.}$$