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\begin{problem}[5.16]  \textit{Compute}$\int_{-\infty}^{\infty}\frac{dx}{x^{4}+1}$  \end{problem}  Let $\sigma_R$ be the semicircle, oriented counterclockwise, counter-clockwise,  formed by the segment $[-R,R]$ on the real axis and the circular arc, $\gamma_R$, of radius $R$ in the upper half of the complex plane. plane.Let $\gamma_1$ be the quarter circle, oriented counter-clockwise, whose path is given by the real and imaginary axes and the portion of $\gamma_R$ in the first quadrant. Let $\gamma_2$ be the quarter circle, oriented counter-clockwise, whose boundary is the real and imaginary axes and the portion of $\gamma_R$ in the second quadrant.