Definite Integral to Infinity of Exponential of -a x by Sine of b x over x

Theorem

 * $\displaystyle \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x = \map \arctan {\frac b a}$

where $a$ and $b$ are real number with $a > 0$.

Proof
Take $a$ constant and define:


 * $\displaystyle \map I b = \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x$

We have:

so:

for some constant $C \in \R$.

We have:

on the other hand:

so:


 * $C = 0$

So we have:


 * $\displaystyle \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x = \map \arctan {\frac b a}$

as required.