Question

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

Answer 1

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

$=\int \frac{dx}{2\left(\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}\right)+\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}+3}$

$=\int \frac{1+{\tan }^2\frac{x}{2}}{4\tan \frac{x}{2}+1-{\tan }^2\frac{x}{2}+3+3{\tan }^2\frac{x}{2}}dx$

$=\int \frac{{\sec }^2\frac{x}{2}}{2{\tan }^2\frac{x}{2}+4\tan \frac{x}{2}+4}dx$

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

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

$=\int \frac{dt}{t^2+2t+2}$

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

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

$={\tan }^{-1}(t+1)+c$

$={\tan }^{-1}(\tan \frac{x}{2}+1)+c$ where $t=\tan \frac{x}{2}$