Definite Integral to Infinity of Exponential of -a x by One minus Cosine x over x Squared

Theorem

 * $\displaystyle \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x = \arccot a - \frac a 2 \map \ln {a^2 + 1} + a \ln a$

where $a$ is a positive real number.

Proof
Set:


 * $\displaystyle \map I a = \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x$

for $a > 0$.

We have:

We then have, from Primitive of $\dfrac 1 x$ and Primitive of $\dfrac x {x^2 + a^2}$:


 * $\displaystyle \map {I'} a = \ln a - \frac 1 2 \map \ln {a^2 + 1} + C$

for some constant $C \in \R$.

Note that:

on the other hand:

so:


 * $\displaystyle \map {I'} a = \ln a - \frac 1 2 \map \ln {a^2 + 1}$

We have:

for some constant $C_2 \in \R$.

Similarly we have:

on the other hand:

We hence have: