Evaluate the following integrals:
$\int \frac{x}{\sqrt{x^{2}+a^{2}}+\sqrt{x^{2}-a^{2}}} d x$
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 $t=x^{2}$
$\Rightarrow \frac{1}{4 a^{2}}\left(\frac{2}{3}\left(x^{2}+a^{2}\right)^{\frac{3}{2}}-\frac{2}{3}\left(x^{2}-a^{2}\right)^{\frac{3}{2}}\right)$
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