Need help solving $\int x \sqrt{\frac {a^2 - x^2} { a^2 + x^2 }} dx$ I have a complicated integral to solve. Can someone provide a better way to solve it than what i did - dividing by a inside the root, and then putting $ t = x / a $, and then putting $t^2 = \cos \theta $ and so many other substitutions.
$$
  \int x \sqrt{\frac {a^2 - x^2} { a^2 + x^2 }} dx
$$

Let $x^2=a^2\cos(y)$. We then have $2xdx = -a^2 \sin(y) dy$. The integral then becomes
\begin{align}
I & = \int x \sqrt{\dfrac{a^2-x^2}{a^2+x^2}} dx = -\int \dfrac{a^2}2 \sin(y) \sqrt{\dfrac{1-\cos(y)}{1+\cos(y)}}dy = -\dfrac{a^2}2 \int \sin(y) \tan(y/2) dy\\
& = -a^2\int \sin^2(y/2)dy
\end{align}
I trust you can finish it from here.
