Question

Solve
$\int \sec \left(\frac{x}{2}\right)dx$.

Answer 1

We have,

$\int \sec \left(\frac{x}{2}\right)dx$

$=\int \sec \left(\frac{x}{2}\right)\cdot \frac{\sec \left(x/2\right)+\tan \left(x/2\right)}{\sec \left(x/2\right)+\tan \left(x/2\right)}dx$

Let $u=\sec \left(x/2\right)+\tan \left(x/2\right)$

$du=\frac{1}{2}\sec \left(\frac{x}{2}\right)\cdot \tan \left(\frac{x}{2}\right)+\frac{1}{2}{\sec }^2\left(\frac{x}{2}\right)dx$

$2du=\sec \left(\frac{x}{2}\right)\tan \left(\frac{x}{2}\right)+{\sec }^2\left(\frac{x}{2}\right)dx$

$\therefore =\int \frac{{\sec }^2\left(x/2\right)+\sec \left(x/2\right)\tan \left(x/2\right)}{\sec \left(\frac{x}{2}\right)+\tan \left(\frac{x}{2}\right)}dx$

$=2\int \frac{du}{u}$

$=2|u|+c$

$=2|\sec \left(\frac{x}{2}\right)+\tan \left(\frac{x}{2}\right)|+c$

Hence, this is answer.