Frullani type integral

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 $$\int dt g(t)$$

HallsofIvy
Homework Helper
Did you mean to say
$$G(t)= \int dt g(t)$$
$$\frac{f(b)- f(a)}{g(b)- g(a)}= \frac{f'(c)}{g'(c)}$$