Frullani's Integral

If $f'(x)$ is continuous and the integral converges,

$$\int_0^\infty {f(ax)-f(bx)\over x}\, dx = [f(0)-f(\infty )]\ln\left({b\over a}\right).$$

References

Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968.