Question

Integrate $\frac{1}{3+cosx}$ with respect to x .

Answer 1

$\int \frac{1}{(3+\cos x)}dx$

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

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

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

$\Rightarrow \int \frac{1}{(3+\frac{1-t^2}{1+t^2})}\cdot \frac{2}{1+t^2}dt$

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

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

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

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

$\Rightarrow \int \frac{1}{2+{\left(\frac{t}{\sqrt{3}}\right)}^2}$

$u=\frac{t}{\sqrt{3}}$;

$du=\frac{1}{\sqrt{3}}dt$

$dt=\sqrt{3}du$

$\Rightarrow \int \frac{1}{2+u^2}\sqrt{3}du$

$\Rightarrow \sqrt{3}\int \frac{1}{2+u^2}du$

$\Rightarrow \sqrt{3}arc\tan \frac{u}{2}+c$

$\Rightarrow \sqrt{3}arc\tan \left(\frac{t/\sqrt{3}}{2}\right)+c$

$\Rightarrow \sqrt{3}arc\tan \left(\frac{t}{2\sqrt{3}}\right)+c$

$\Rightarrow \sqrt{3}arc\tan \frac{1}{2\sqrt{3}}\tan \left(\frac{x}{2}\right)+c$