Solve an integral by completeing a square

Question

By completing the square, I have to find:

$$\int \frac{x^2}{\sqrt{2x-x^2}} dx$$

So far, I have the integral in the form:

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

I have no idea where to go from here, or if what I'm doing is even correct. How can this integral be solved.

Notes: I'm a bit rusty on my calculus.

Answer 1

Hint:

Substitutions $u=1-x$, then $u=\sin\theta$ or $\cos\theta$. You'll get the integral of a trigonometric polynomial.