# Frullani type integral

by mhill
Tags: frullani, integral, type
 P: 193 under what condition the does the equality hold ? $$\int_{0}^{\infty} dt \frac{ f(at)-f(bt)}{g(t)}= (G(b)-G(a))(f(0)-f(\infty))$$ and $$\int dt g(t)$$
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,706 Did you mean to say $$G(t)= \int dt g(t)$$ in your last line?
 P: 193 yes i meant the integral of g(t) ,i have been looking for generalizations of this integral but without any luck , under that condition can define g(t) and f(at) f(bt) so the integral above converge ?? thank
Math
Emeritus
Thanks
PF Gold
P: 38,706

## Frullani type integral

The generalized mean value theorem says that if f and g are differentiable on (a, b) and continuous on [a, b], then there exist c in (a, b) such that
$$\frac{f(b)- f(a)}{g(b)- g(a)}= \frac{f'(c)}{g'(c)}$$

That looks like an integral version to me.

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