Frullani's Integral

Theorem
Let $a, b > 0$.

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

Suppose that $\displaystyle f \left({\infty}\right) = \lim_{x \to \infty} f \left({x}\right)$ exists, and is finite.

Then:
 * $\displaystyle \int_0^\infty \frac {f \left({a x}\right) - f \left({b x}\right)} x \rd x = \left({f \left({\infty}\right) - f \left({0}\right)}\right) \ln \frac a b$