Question:
Evaluate: $\int \cos ^{2} \frac{\mathrm{x}}{2} \mathrm{dx}$
Solution:
We know, $\cos ^{2} x=\frac{1+\cos 2 x}{2}$
$\therefore$ The given equation becomes,
$\Rightarrow \int \frac{1+\cos 2 \frac{x}{2}}{2} d x=\int \frac{1+\cos x}{2} d x$
We know $\int \cos a x d x=\frac{1}{a} \sin a x+c$
$\Rightarrow \frac{1}{2} \int \mathrm{dx}+\frac{1}{2} \int \cos (\mathrm{x}) \mathrm{dx}$
$\Rightarrow \frac{x}{2}+\frac{1}{2} \sin (x)+c$
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