Integration using substitution: $\int \frac{1}{\sqrt{(1 - x^2)}\arcsin^3x} dx $

Question

I am trying to solve this integration:

$$\int \frac{1}{\sqrt{(1 - x^2)}\arcsin^3x} dx $$

using substitution. My problem is that I can not find the right way to use substitution in this case.

The result should be:

$$ C- \frac{1}{2(\arcsin^2x)}$$

note: I guess that I should use this formula:

$$ \int \frac{1}{\sqrt{1-x^2}}dx = \arcsin x + C $$

Thank you.

Answer 1

Notice that the derivative of $\arcsin(x)$ appears in this integrand. In general, this tells us that $u$-substitution would be a good thing to try; in our case, if we let $u = \arcsin(x)$, we get $\displaystyle du = \frac{dx}{\sqrt{1 - x^2}}$. Thus our integral becomes:

$$\int \frac{1}{\sqrt{1 - x^2} \ \arcsin^3(x)} \ dx \ =\ \int \frac{1}{u^3} \ du$$