How is this integral solved?

Question

$$\int \sqrt{x-x^2}dx= \int\frac{x-x^2}{\sqrt{x-x^2}}dx=(Ax+B)\sqrt{x-x^2}+k \int\frac{1}{\sqrt{x-x^2}}dx$$ after this some term is gotten with A,B,C and $x$ then these coefficients, that are beside $x,x^2$ are equalized somehow, this I do not understand, can anyone provide some assistance on how this is done ?

Answer 1

$\sqrt{x-x^2} = \sqrt{-\left(x-\dfrac{1}{2}\right)^2 + \left(\dfrac{1}{2}\right)^2}$. Now let $x-\dfrac{1}{2}=\dfrac{\sin \theta}{2}$. Can you manage to continue?