<s>[INST] How to integrate $\int x\sqrt { \frac {a^2-x^2}{a^2+x^2} } \, dx$? How to integrate $\displaystyle\int  x\sqrt {\frac{a^2 - x^2}{a^2+x^2}} \,  dx$ ?
I tried this for quite some time, but not being able to solve. Help!
 [/INST] Hint:
$$x\sqrt{\frac{a^2-x^2}{a^2+x^2}}=x\sqrt{\frac{1-(\frac{x}{a})^2}{1+(\frac{x}{a})^2}}$$
and use the trigonometric identity
$$\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=|\tan\left(\frac{\theta}{2}\right)|$$
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