If f(x) = sinx + cosx and g(x) = $\left\{\begin{array}{l}\frac{\left|\mathrm{x}\right|}{\mathrm{x}},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}\ne 0\\ 2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}=0\end{array}\right.$ then the value of ${\int }_{-\mathrm{\pi }/4}^{2\mathrm{\pi }} \mathrm{gof}\left(\mathrm{x}\right)\mathrm{dx}=$

$$\begin{gathered} \mathrm{g}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\frac{\mathrm{x}}{\mathrm{x}}=1,\mathrm{x}>0\\ 2,\mathrm{x}=0\\ \frac{-\mathrm{x}}{\mathrm{x}}=-1,\mathrm{x}<0\end{array};\mathrm{g}\left(\mathrm{f}\right(\mathrm{x}\left)\right)=\left\{\begin{array}{l}1,\sin \mathrm{x}+\cos \mathrm{x}>0\\ 2,\sin \mathrm{x}+\cos \mathrm{x}=0\\ -1,\sin \mathrm{x}+\cos \mathrm{x}<0\end{array}\right.\right. \\ \text{ From }\frac{-\mathrm{\pi }}{4}\mathrm{to}\frac{3\mathrm{\pi }}{4},\sin \mathrm{x}+\cos \mathrm{x}>0 \\ \text{ From }\frac{3\mathrm{\pi }}{4}\text{ to }\frac{7\mathrm{\pi }}{4},\sin \mathrm{x}+\cos \mathrm{x}<0;\text{ From } \\ \frac{7\mathrm{\pi }}{4}\text{ to }2\mathrm{\pi },\sin \mathrm{x}+\cos \mathrm{x}>0 \\ \therefore {\int }_{-\frac{\mathrm{\pi }}{4}}^{2\mathrm{\pi }/4} \mathrm{g}\left(\mathrm{f}\right(\mathrm{x}\left)\right)\mathrm{dx}={\int }_{-\mathrm{\pi }/4}^{3\mathrm{\pi }/4} 1\mathrm{dx}+{\int }_{3\mathrm{\pi }/4}^{7\mathrm{\pi }/4} -1\mathrm{dx}+{\int }_{7\mathrm{\pi }/4}^{2\mathrm{\pi }} 1\mathrm{dx}=\frac{\mathrm{\pi }}{4} \end{gathered}$$