TITLE: $\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$ Evaluate Integral
QUESTION [28 upvotes]: Compute the following integral:
$$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$$
Any hint, suggestion is welcome.

REPLY [3 votes]: $$\int_0^{\infty} \frac{e^{-x}\sin x}{x}\,dx=\int_0^{\infty}\int_0^{\infty} e^{-x}\sin x \,e^{-xy}\,dy\,dx=\int_0^{\infty} \int_0^{\infty} e^{-x(1+y)}\sin x\,dx\,dy$$
From integration by parts or otherwise, one can show that:
$$\int_0^{\infty}e^{-x(1+y)}\sin x\,dx=\frac{1}{1+(1+y)^2}$$
Hence
$$\int_0^{\infty} \int_0^{\infty} e^{-x(1+y)}\sin x\,dx\,dy=\int_0^{\infty} \frac{dy}{1+(1+y)^2}=\left(\arctan(1+y)\right|_0^{\infty}=\boxed{\dfrac{\pi}{4}}$$