Question:
Evaluate $\int \frac{1}{\left(x^{2}+2\right)\left(x^{2}+5\right)} d x$
Solution:
$\int \frac{d x}{\left(x^{2}+5\right)\left(x^{2}+2\right)}$
By partial fractions, $\frac{1}{\left(x^{2}+5\right)\left(x^{2}+2\right)}=\frac{A}{x^{2}+5}+\frac{B}{x^{2}+2}$
Solving these two equations, $2 A+5 B=1$ and $A+B=0$
We get $A=-1 / 3$ and $B=1 / 3$
$=-\frac{1}{3} \int \frac{d x}{\left(x^{2}+5\right)}+\frac{1}{3} \int \frac{d x}{\left(x^{2}+2\right)}=-\frac{1}{3} \cdot \frac{1}{\sqrt{5}} \tan ^{-1} \frac{x}{\sqrt{5}}+\frac{1}{3} \cdot \frac{1}{\sqrt{2}} \tan ^{-1} \frac{x}{\sqrt{2}}+c$