Question

Evaluate the following integrals:
$\underset{0}{\overset{\mathrm{\pi }/2}{\int }}\left|\cos 2x\right|dx$

Answer 1

$$\begin{gathered} {\int }_0^{\frac{\mathrm{\pi }}{2}}\left|\cos 2x\right|\mathit{}\mathit{d}x \\ \mathrm{We}\mathrm{know}\mathrm{that},\left|\cos 2x\right|=\left\{\begin{array}{l}-\cos 2x,\frac{\mathrm{\pi }}{4}\le x\le \frac{\mathrm{\pi }}{2}\\ \cos 2x,0<x\le \frac{\mathrm{\pi }}{4}\end{array}\right. \\ \therefore I={\int }_{-2}^2\left|\cos 2x\right|\mathit{d}x \\ \Rightarrow I={\int }_0^{\frac{\mathrm{\pi }}{4}}\cos 2xdx-{\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}\cos 2xdx \\ \Rightarrow I={\left[\frac{\sin 2x}{2}\right]}_0^{\frac{\mathrm{\pi }}{4}}-{\left[\frac{\sin 2x}{2}\right]}_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}} \\ \Rightarrow I=\frac{1}{2}-0-0+\frac{1}{2} \\ \Rightarrow I=1 \end{gathered}$$