Question

Evaluate the following integrals:
$\underset{-\mathrm{\pi }/4}{\overset{\mathrm{\pi }/4}{\int }}\left|\sin x\right|dx$

Answer 1

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