Integration of Root Terms

Question

I was wondering if x (root(1-x^2)dx can be solved as:

root(x^2)root(1-x^2)dx

root(x^2(1-x^2)) dx

root(x^2 - x^4) dx

root(-x^2)dx

If this is correct, I'm stuck at the final expression. If not, how do we solve this?

Answer 1

Note that $\int \sqrt{x^2-x^4} dx \neq \int \sqrt{-x^2}$, as $x^2$ and $x^4$ have different exponents. Instead, consider $u=x^2$.