How do I integrate $\int x\sqrt{A^2 - x^2} \mathrm dx$ $$ \int x\sqrt{A^2 - x^2} \mathrm dx $$
I never dealt with 2 variables on integration, what should I do here?

Alternative way is using the trigonometric substitution $x=A\sin{t}.$
Then $$dx=A\cos{t}\;dt,\\
{A^2 - x^2}=A^2-A^2{\sin^2{t}}=A^2{\cos^2{t}},$$ therefore
$$\int x\sqrt{A^2 - x^2}\;  dx = \int {A\sin{t}\,A{\cos{t}}}A\cos{t}\;dt =\\
= -A^3 \int {\cos^2{t}\;d(\cos{t})}=-\frac{A^3}{3}\cos^3{t}+C=  -\frac{A^3}{3} \left(1-\sin^2{t} \right)^{\tfrac{3}{2}}+C =\\
=-\frac{A^3}{3}\left({1-\frac{x^2}{A^2}}\right)^{\tfrac{3}{2}}+C=-\frac{1}{3}\left({A^2-{x^2}}\right)^{\tfrac{3}{2}}+C.$$
