# 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: 39,682 Did you mean to say $$G(t)= \int dt g(t)$$ in your last line?
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 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.