Question

Evaluate $\stackrel{\frac{\pi }{2}}{\underset{-\frac{\pi }{2}}{\int }}\sin |x|dx$.

Answer 1

${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sin \left|x\right|dx$

$=-{\int }_{-\frac{\pi }{2}}^0\sin xdx+{\int }_0^{\frac{\pi }{2}}\sin xdx$

$=-{[-\cos x]}_{-\frac{\pi }{2}}^0+{[-\cos x]}_0^{\frac{\pi }{2}}$

$=[\cos 0-\cos \frac{\pi }{2}]-[\cos \frac{\pi }{2}-\cos 0]$

$=(1-0)-(0-1)$

$=2$

Hence,

${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sin \left|x\right|dx=2$