How should I calculate this integral
$$\int\limits_{-\infty}^\infty\frac{\sin x}{x(1+x^2)}\,dx\quad?$$
I have tried forming an indented semicircle in the upper half complex plane using the residue theorem and I tried to integrate along a curve that went around the complex plane and circled the positive real axis (since the integrand is even). Nothing has worked out for me. Please help!
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