Evaluate the following integrals:

Rationlize the given equation we get

$\Rightarrow \int \frac{x}{\sqrt{x^{2}+a^{2}}+\sqrt{x^{2}-a^{2}}} \times \frac{\sqrt{x^{2}+a^{2}}-\sqrt{x^{2}-a^{2}}}{\sqrt{x^{2}+a^{2}}-\sqrt{x^{2}-a^{2}}} d x$

$\Rightarrow \int \frac{x\left(\sqrt{x^{2}+a^{2}}-\sqrt{x^{2}-a^{2}}\right)}{2 a^{2}} d x$

Assume $\mathrm{x}^{2}=\mathrm{t}$

$2 x \cdot d x=d t$

$\Rightarrow \mathrm{dx}=\frac{\mathrm{dt}}{2 \mathrm{x}}$

Substituting $\mathrm{t}$ and $\mathrm{dt}$

$\Rightarrow \int \frac{\left(\sqrt{t+a^{2}}-\sqrt{t-a^{2}}\right)}{4 a^{2}} d t$

$\Rightarrow \frac{1}{4 a^{2}} \int\left(\sqrt{t+a^{2}}-\sqrt{t-a^{2}}\right) d t$

$\Rightarrow \frac{1}{4 a^{2}} \int\left(t+a^{2}\right)^{1 \backslash 2} d t-\int\left(t-a^{2}\right)^{1 / 2} d t$

$\Rightarrow \frac{1}{4 a^{2}}\left(\frac{2}{3}\left(t+a^{2}\right)^{\frac{3}{2}}-\frac{2}{3}\left(t-a^{2}\right)^{\frac{3}{2}}\right)$

But $t=x^{2}$

$\Rightarrow \frac{1}{4 a^{2}}\left(\frac{2}{3}\left(x^{2}+a^{2}\right)^{\frac{3}{2}}-\frac{2}{3}\left(x^{2}-a^{2}\right)^{\frac{3}{2}}\right)$