$\int \frac{\mathrm{dx}}{{\mathrm{a}}^2{\sin }^2\mathrm{x}+{\mathrm{b}}^2{\cos }^2\mathrm{x}}$ is equal to 

Solution:

$\int \frac{\mathrm{dx}}{{\mathrm{a}}^2{\sin }^2\mathrm{x}+{\mathrm{b}}^2{\cos }^2\mathrm{x}}=\int \frac{{\sec }^2\mathrm{xdx}}{{\mathrm{a}}^2{\tan }^2\mathrm{x}+{\mathrm{b}}^2}=\frac{1}{{\mathrm{a}}^2}\int \frac{\mathrm{dt}}{{\mathrm{t}}^2+{\left(\frac{\mathrm{b}}{\mathrm{a}}\right)}^2}$

$\mathrm{where}, \begin{array}{r}\mathrm{t}=\tan \mathrm{x}=\frac{1}{{\mathrm{a}}^2}\cdot \frac{\mathrm{a}}{\mathrm{b}}{\tan }^{-1}\frac{\mathrm{t}}{\mathrm{b}/\mathrm{a}}+\mathrm{C}\\ =\frac{1}{\mathrm{ab}}{\tan }^{-1}\frac{\mathrm{a}(\tan \mathrm{x})}{\mathrm{b}}+\mathrm{C}\end{array}$