A Mathematical Proof for an Improper Integral

Question

I got the following solution for the integral $I$ from Wolfram and I have verified the solution numerically which seems to be correct! Does someone have an idea about the mathematical proof?

$I = \int_{0}^{\infty} \frac{e^{-ax}sin(bx)}{x} dx = \arctan(\frac{b}{a})$

in which a and b are some positive constants.

Thanks in advance.

Answer 1

Start with $\int_0^\infty e^{-ax} \cos(bx) \, dx$, which is easily worked out. Then integrate with respect to $b$. Use $b=0$ to establish the integration constant. Wave your hands to explain the validity of exchanging the two integrals (or cite Fubini's Theorem).

Answer 2

You can follow the steps

i) Differentiate $I$ w.r.t. $a$.

ii) evaluate the integral $I_a$.

iii) integrate with respect with $a$ and note that $\lim_{a\to \infty} I =0. $

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