Q12 (b)

$\int\frac{1}{2+\cos x}\,dx=\frac{2}{\sqrt3}\tan^{-1}\left(\frac{\tan(x/2)}{\sqrt3}\right)+C$

Working Out:
$\quad \int\frac{1}{2+\cos x}\,dx$

$t=\tan\left(\frac{x}{2}\right)$

$\frac{dt}{dx}=\frac{1}{2}\sec^2\left(\frac{x}{2}\right)=\frac{1+t^2}{2}$

$\therefore dx=\frac{2}{1+t^2}\,dt$

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

$=\int\frac{2}{3+t^2}\,dt$

$=\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{t}{\sqrt{3}}\right)+C$

$=\frac{2}{\sqrt{3}}\tan^{-1}\left[\frac{\tan\left(\frac{x}{2}\right)}{\sqrt{3}}\right]+C$