Integrating $\frac{a^x-b^x}{x}dx$

Question

I'm having problems solving this integral:$$\int_{0}^{\infty} \frac{a^x-b^x}{x}dx$$ Could this be consider a Frullani integral? so far I tried with an $x=e^{-t}$ substitution but I'm blocked.

Answer 1

It is indeed a Frullani integral. Write the integrand as $$\int_0^\infty \frac{e^{x\ln(a)}-e^{x\ln(b)}}{x}dx$$ Then, using the formula that Frullani provided for integrals of this sort, we have $$\int_0^\infty \frac{a^x-b^x}{x}dx=\ln\frac{\ln(b)}{\ln(a)}$$ which converges for $a,b$ between $0$ and $1$.