`int_0^a xsqrt((a^2-x^2)/(a^2+x^2))dx`

Explanation

`
\begin{array}{l}
\quad \text { Put } x^{2}=t \\
=\int \sqrt{\frac{a^{2}-t}{a^{2}+t}} \frac{d t}{2} \\
=\frac{1}{2} \int \sqrt{\frac{a^{2}-t}{a^{2}+t} \times \frac{a^{2}-t}{a^{2}-t}} d t \\
=\frac{1}{2} \int \frac{a^{2}-t}{\sqrt{a^{4}-t^{2}}} d t \\
=\frac{1}{2} \int \frac{a^{2}}{\sqrt{a^{4}-t^{2}}} d t-\frac{1}{2} \int \frac{t d t}{\sqrt{a^{4}-t^{2}}} \\
=\frac{a^{2}}{2} \sin ^{-1}\left(\frac{t}{a^{2}}\right)+\frac{1}{4} \int \frac{d u}{\sqrt{u}} \\
...