Explanation

(a)
Let $I = \int_{}^{}\frac{1}{1 + \cos x + \sin x}\ dx$
$= \int_{}^{}\frac{1}{1 + \frac{1 - \tan^{2}\frac{x}{2}}{1 + \tan^{2}\frac{x}{2}} + \frac{2\tan\frac{x}{2}}{1 + \tan^{2}\frac{x}{2}}}\ dx$
$= \int_{}^{}\frac{\sec^{2}\frac{x}{2}}{2\left( 1 + \tan\frac{x}{2} \right)}\ dx$
Put $\tan\frac{x}{2} = t$
$\Rightarrow \frac{1}{2}\sec^{2}\frac{x}{2}\ dx = dt$
$\therefore\ I = \int_{}^{}\frac{dt}{1 + t}\ = \log{(1 + t)} + c$
$= \log\left| 1 + \tan\frac{x}{2} \right| + c$