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!
Best Answer
$$ \int_{-\infty}^\infty\frac{\sin x}{x(1+x^2)}\mathrm dx=\int_{-\infty}^\infty\frac{\Im\mathrm e^{\mathrm ix}}{x(1+x^2)}\mathrm dx=\int_{-\infty}^\infty\frac{\Im\left(\mathrm e^{\mathrm ix}-1\right)}{x(1+x^2)}\mathrm dx=\Im\int_{-\infty}^\infty\frac{\mathrm e^{\mathrm ix}-1}{x(1+x^2)}\mathrm dx\;. $$