Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Secant of p x

Theorem

 * $\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \frac 1 2 \map \ln {\frac {b^2 + p^2} {a^2 + p^2} }$

where:
 * $a$ and $b$ are non-negative real numbers
 * $p$ is a real number.

Proof
Fix $p$ and set:


 * $\ds \map I \alpha = \int_0^\infty \frac {e^{-\alpha x} } {x \sec p x} \rd x$

for all $\alpha \ge 0$.

Then:


 * $\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \map I a - \map I b$

We have:

so:

for all $\alpha \ge 0$, for some constant $C \in \R$.

We then have: