Definite Integral to Infinity of Arctangent of p x minus Arctangent of q x over x

Theorem

 * $\displaystyle \int_0^\infty \frac {\arctan p x - \arctan q x} x \rd x = \frac \pi 2 \ln \frac p q$

where $p$ and $q$ are strictly positive real numbers.

Proof
Note that the integrand is of the form:


 * $\displaystyle \int_0^\infty \frac {\map f {p x} - \map f {q x} } x \rd x$

where:


 * $\map f x = \arctan x$

We have, by Derivative of Arctangent Function:


 * $\map {f'} x = \dfrac 1 {1 + x^2}$

which is continuous on $\R$.

By Limit to Infinity of Arctangent Function:


 * $\displaystyle \lim_{x \mathop \to \infty} \map f x = \lim_{x \mathop \to \infty} \arctan x = \frac \pi 2$

As $f$ is continuously differentiable and $\displaystyle \lim_{x \mathop \to \infty} \map f x$ exists and is finite, we may apply Frullani's Integral, giving: