Question:
Evaluate the following integral:
$\int \frac{x}{\left(x^{2}+4\right) \sqrt{x^{2}+1}} d x$
Solution:
assume $x^{2}+1=u^{2}$
$x d x=u d u$
$\int \frac{u d u}{\left(u^{2}+3\right) u}$
$\int \frac{d u}{\left(u^{2}+3\right)}$
Using identity $\int \frac{1}{x^{2}+1} d x=\arctan (x)$
$\frac{1}{\sqrt{3}} \arctan \left(\frac{\mathrm{u}}{\sqrt{3}}\right)+\mathrm{c}$
Substituting $u=\sqrt{1+x^{2}}$
$\frac{1}{\sqrt{3}} \arctan \left(\frac{\sqrt{1+x^{2}}}{\sqrt{3}}\right)+c$