Question

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left|\cos \frac{x}{2}\right|dx=$

• A. $1$
• B. $-2$
• C. $\sqrt{2}$
• D. $0$

Answer 1

The correct option is C

$\sqrt{2}$

Explanation for the Correct Answer:

Finding the value of the given integration,

$\begin{array}{rcl}{\int }_0^{\frac{\mathrm{\pi }}{2}}\left|\cos \frac{x}{2}\right|dx& =& {\int }_0^{\frac{\mathrm{\pi }}{2}}\cos \frac{x}{2}dx\left[\left|\cos x\right|=\cos x;-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}\right]\\ & =& {\left[\frac{\sin \frac{x}{2}}{{\displaystyle \frac{1}{2}}}\right]}_0^{\frac{\mathrm{\pi }}{2}}\\ & =& 2\left[\sin \frac{\mathrm{\pi }}{4}-\sin 0\right]\\ & =& 2\left[\frac{1}{\sqrt{2}}-0\right]\\ & =& \sqrt{2}\end{array}$

Hence, option (C) is the correct answer.