The value of the integral ${\int }_0^{n\pi +t} \left(\right|\cos x|$ + |sin x|) dx is 

Since the period of $|\sin x|+|\cos x$ is $\pi /2$

$\begin{array}{l}{\int }_0^{n\pi +t} \left(\right|\sin x|+|\cos x\left|\right)\mathrm{d}x=2n{\int }_0^{\pi /2} \left(\right|\sin x|+|\cos x\left|\right)\mathrm{d}x+\\ {\int }_0^t \left(\right|\sin x|+|\cos x\left|\right)\mathrm{d}x\\ =2n{\int }_0^{\pi /2} (\sin x+\cos x)\mathrm{d}x+{\int }_0^t (\sin x+\cos x)\mathrm{d}x\\ =\left(2n\right)\left(2\right)+\sin t-\cos t+1\\ =(4n+1)+\sin t-\cos t.\end{array}$