Question

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

Answer 1

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

Substitute $t=\tan \frac{x}{2}⟹x=2{\tan }^{-1}\left(t\right)$

$\therefore dx=\frac{2}{1+t^2}dt$ and $\sin x=\frac{2t}{1+t^2},\cos x=\frac{1-t^2}{1+t^2}$

$\therefore I=\int \frac{\frac{2}{1+t^2}}{\frac{4t}{1+t^2}+\frac{1-t^2}{1+t^2}+3}dt$

$\therefore I=\int \frac{2}{4t+1-t^2+3+3t^2}dt$

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

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

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

$\therefore I={\tan }^{-1}(t+1)+c$ (as $\int \frac{1}{x^2+1}dx={\tan }^{-1}x$)

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

This is the required solution.