Frullani's Integral

Theorem
Let $a, b > 0$.

Let $f$ be a function continuously differentiable on the non-negative real numbers.

Suppose that $\displaystyle \map f \infty = \lim_{x \mathop\to \infty} \map f x$ exists, and is finite.

Then:
 * $\displaystyle \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x = \paren {\map f \infty - \map f 0} \ln \frac a b$