Evaluate the following integrals:

In the given equation

$\cos x=\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}$

Also, $\cos ^{2} \frac{x}{2}+\sin ^{2} \frac{x}{2}=1$

Substituting in the above equation we get,

$\Rightarrow \int \frac{1}{\sqrt{\cos ^{2} \frac{x}{2}+\sin ^{2} \frac{x}{2}+\left(\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\right)}} d x$

$\Rightarrow \int \frac{1}{\sqrt{\cos ^{2} \frac{x}{2}+\sin ^{2} \frac{x}{2}+\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}}} d x$

$\Rightarrow \int \frac{1}{\sqrt{2 \cos ^{2} \frac{x}{2}}} d x$

$\Rightarrow \int \frac{1}{\sqrt{2} \cos \frac{x}{2}} d x$

$\Rightarrow \frac{1}{\sqrt{2}} \int \sec \frac{x}{2} d x$

$\Rightarrow \frac{1}{\sqrt{2}} \ln \left|\sec \frac{x}{2}+\tan \frac{x}{2}\right|+c$