Question

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

Answer 1

$$\begin{gathered} {\int }_{\frac{-\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left\{\sin \left|x\right|+\cos \left|x\right|\right\}\mathit{d}x \\ \mathrm{Since},f\left(-x\right)=\sin \left|-x\right|+\cos \left|-x\right|=\sin \left|x\right|+\cos \left|x\right|=f\left(x\right) \\ So,f\left(x\right)\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{function}. \\ \therefore I=2{\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\sin x+\cos x\right)\mathrm{dx} \\ \Rightarrow I=2{\left[-\cos x+\sin x\right]}_0^{\frac{\mathrm{\pi }}{2}} \\ \Rightarrow I=2\left(0+1+1-0\right) \\ \Rightarrow I=4 \end{gathered}$$