Evaluate the following integrals:

take out $a^{2}$

$=\frac{1}{a^{2}} \int \frac{1}{x^{2}-\frac{b^{2}}{a^{2}}} d x$

$=\frac{1}{a^{2}} \int \frac{1}{x^{2}-\left(\frac{b}{a}\right)^{2}} d x=\frac{1}{a^{2}} * \frac{1}{2\left(\frac{b}{a}\right)} \log \left[\frac{x-\left(\frac{b}{a}\right)}{x+\frac{b}{a}}\right]+c\left\{\operatorname{since} \int \frac{1}{a^{2}-x^{2}} d x=\frac{1}{2 a} \log \frac{x+a}{x-a}+c\right\}$

$=\frac{1}{2 a b} \log \frac{a x-b}{a x+b}+c$