# Evaluate the following integral:

Question:

Evaluate the following integral:

$\int \frac{1}{\left(2 x^{2}+3\right) \sqrt{x^{2}-4}} d x$

Solution:

assume $x=\frac{1}{t}$

$\mathrm{dx}=-\frac{1}{\mathrm{t}^{2}} \mathrm{dt}$

$-\int \frac{t d t}{\left(3 t^{2}+2\right)\left(\sqrt{1-4 t^{2}}\right.}$

Assume $1-4 t^{2}=u^{2}$

$-4 t d t=u d u$

$-\frac{1}{4} \int \frac{\text { udu }}{\left(\frac{11-3 u^{2}}{4}\right) u}$

$-\frac{1}{3} \int \frac{d u}{\left(\frac{11}{3}-u^{2}\right)}$

Using identity $\int \frac{d z}{(z)^{2}-1}=\frac{1}{2} \log \left|\frac{z-1}{z+1}\right|+c$

$\frac{1}{2 \sqrt{33}} \log \left|\frac{u-\sqrt{\frac{11}{3}}}{u+\sqrt{\frac{11}{3}}}\right|+c$

Substituting $u=\sqrt{1-4 t^{2}}$

$\frac{1}{2 \sqrt{33}} \log \left|\frac{\sqrt{1-4 t^{2}}-\sqrt{\frac{11}{3}}}{\sqrt{1-4 t^{2}}+\sqrt{\frac{11}{3}}}\right|+c$

Substituting $\mathrm{t}=\frac{1}{\mathrm{x}}$

$\frac{1}{2 \sqrt{33}} \log \left|\frac{\sqrt{1-\frac{4}{x^{2}}}-\sqrt{\frac{11}{3}}}{\sqrt{1-\frac{4}{x^{2}}}+\sqrt{\frac{11}{3}}}\right|+c$