Question

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

Answer 1

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

Substitute $\sin x=\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$ and $\cos x=\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$

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

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

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

$=\int \frac{{\sec }^2\frac{x}{2}dx}{2[{(\tan \frac{x}{2}+1)}^2+1]}$

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

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

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

$={\tan }^{-1}(\tan \frac{x}{2}+1)+c$ where $c$ is the constant of integration.