Question

Evaluate $\int \frac{dx}{(2\sin x+\cos x+3)}$=

Answer 1

$\int \frac{1}{2\sin x+\cos x+3}dx$

We know that $\sin x=\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$

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

$\Rightarrow 1+{\tan }^2\frac{x}{2}={\sec }^2\frac{x}{2}$

Take $\tan \frac{x}{2}=t\Rightarrow \frac{1}{2}{\sec }^2\frac{x}{2}dx=dt$

Substituting these values in the integral we get

$\Rightarrow \int \frac{2}{4t+1-t^2+3+3t^2}dt$

$\Rightarrow \int \frac{2}{2t^2+4t+4}dt$

$\Rightarrow \int \frac{1}{t^2+2t+2}dt$

$\Rightarrow \int \frac{1}{(t+1{)}^2+1}dt$

$\Rightarrow {\tan }^{-1}(t+1)+C$

$\Rightarrow {\tan }^{-1}(\tan \frac{x}{2}+1)+C$