$\underset{0}{\overset{\pi }{\int }}(\cos x+\left|\cos x\right|)dx=$

$I=\underset{0}{\overset{\pi }{\int }}\left(\cos x+\left|\cos x\right|\right) dx$

$=\underset{0}{\overset{\pi /2}{\int }}(\cos x+\cos x)dx+\underset{\pi /2}{\overset{\pi }{\int }}(\cos x-\cos x)dx$

$\left(\because x\in \left(0,\frac{{\displaystyle \pi }}{{\displaystyle 2}}\right)\right)\Rightarrow \cos x>0$

$\Rightarrow \left|\cos x\right|=\cos x and x\in \left(\frac{{\displaystyle \pi }}{{\displaystyle 2}},\pi \right)\Rightarrow \cos x<0\Rightarrow \left|\cos x\right|=-\cos x$

$I=2{\left(\sin x\right)}_0^{\pi /2}+0$

$I=2(1-0)$