Definite Integration — Maths STD 12 Science — Question

$\int_0^{\frac{\pi}{2}} \frac{1}{1+\cos x} d x$
Solving the integral without limits,
$\begin{aligned} & \int \frac{1}{1+\cos x} d x \\ & =\int \frac{1}{2 \cos ^2\left(\frac{x}{2}\right)} d x \\ & =\frac{1}{2} \int \sec ^2\left(\frac{x}{2}\right) d x \\ & =\frac{1}{2}\left[\frac{\tan \left(\frac{x}{2}\right)}{\frac{1}{2}}\right]+C \\ & =\tan \left(\frac{x}{2}\right)+C\end{aligned}$
Substituting the limits,we get
$\begin{aligned} & =\left[\tan \left(\frac{x}{2}\right)\right]_0^{\frac{\pi}{2}} \\ & =\left[\tan \left(\frac{\pi}{4}\right)-\tan 0\right] \\ & =1\end{aligned}$