I want to evaluate the following integral, using complex integration:
$$\int_{0}^{\infty} \frac{\sin x}{x} \mathrm{d}x.$$ I know the solution is $\frac{\pi}{2}$, as given by the exercise.
It is hinted that $$\int_{0}^{\infty} \frac{\sin x}{x} \mathrm{d}x = \frac{1}{2\mathrm{i}} \int_{-\infty}^{\infty}\frac{e^{\mathrm{i}x} - 1}{x}\mathrm{d}x,$$ and that I'm supposed to use the indented semicircle as my toy contour.
Yet, I don't know how to go from here or approach this problem. Can I show the hint by using Cauchy's theorem?