Question

Evaluate: $\int \frac{1}{{\sin }^{2}x+4{\cos }^{2}x}dx$

Answer 1

Now,

$\int \frac{1}{{\sin }^{2}x+4{\cos }^{2}x}dx$

$=\int \frac{{\sec }^{2}x}{{\tan }^{2}x+2^2}dx$ [ Dividing the numerator and denominator by ${\cos }^{2}x$]

$=\int \frac{d\left(\tan x\right)}{{\tan }^{2}x+2^2}$

$=\frac{1}{2}{\tan }^{-1}\frac{\tan x}{2}+c$ [ Where $c$ is integrating constant]

$=\frac{1}{2}{\tan }^{-1}\frac{\tan x}{2}+c$