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Frullani Integrals
Definite integrals of the form


 * $$\int \limits _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x$$
 * where $${f(x)}$$ is a function over $${x\geq 0}$$, and the limit of $${f(x)}$$ exists at $${\infty}$$

are known as Frullani integrals. The following formula for their general solution holds under certain conditions:


 * $${\int \limits _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x=(f(0)-f(\infty))\ln {\frac {b}{a}}}$$.

This can be proved using the method of differentiation under the integral sign when $$f'(x)$$ is continuous.