Evaluate the following integrals:

Taking out $b^{2}, \frac{1}{b^{2}} \int \frac{1}{\left(\frac{a^{2}}{b^{2}}\right)-x^{2}} d x$

$=\frac{1}{\mathrm{~b}^{2}} \int \frac{1}{\left(\frac{\mathrm{a}^{2}}{\mathrm{~b}^{2}}\right)-\mathrm{x}^{2}} \mathrm{dx}=\frac{1}{\mathrm{~b}^{2}} \int \frac{1}{\left(\frac{2}{\mathrm{~b}}\right)^{2}-\mathrm{x}^{2}} \mathrm{dx}$

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

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