The value of ∫ 0 π cos x 3 d x is

$I={\int }_0^{\pi }{\left|\cos x\right|}^{3}dx=2{\int }_0^{\frac{\mathrm{\pi }}{2}}\mathrm{c}\mathrm{o}{\mathrm{s}}^{3}xdx$

$=2{\int }_0^{\frac{\pi }{2}}\left(1-\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}x\right)\mathrm{c}\mathrm{o}\mathrm{s}xdx$

$=2\left[\underset{0}{\overset{\frac{\pi }{2}}{\int }}\mathrm{c}\mathrm{o}\mathrm{s}xdx-\underset{0}{\overset{\frac{\pi }{2}}{\int }}\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}x\mathrm{c}\mathrm{o}\mathrm{s}xdx\right]$

$=2{\left[\mathrm{s}\mathrm{i}\mathrm{n}x-\frac{\mathrm{s}\mathrm{i}{\mathrm{n}}^{3}x}{3}\right]}_0^{\frac{\pi }{2}}$

$=2\left[1-\frac{1}{3}-0+0\right]=\frac{4}{3}$