<s>[INST] Solve: $\int x^3\sqrt{4-x^2}dx$ $$\int x^3\sqrt{4-x^2}dx$$
I tried changing the expression like this:
$$\int x^3\sqrt{2^2-x^2}dx=\int x^3\sqrt{(2-x)(2+x)}dx$$
And that's where I got stuck. I tried integration by parts but it doesn't simplify the expression.
 [/INST] Substituting $x=2\sin t$, we get:
$$ \int x^3\sqrt{2^2-x^2}dx=32\int \sin^3 t \cos^2 t dt=32\int(1-\cos^2 t)\cos ^2 t\sin t dt$$
$$=32\int (\cos^2 t-\cos ^4 t)\sin tdt$$
$$=-32\int z^2 dz + 32\int z^4 dz$$ Where $z=\cos t.$ 
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