Is this a good approximation for $\int_a^b\frac{f(x)}{x}dx$?

Question

I've derived this approximation: $$\int_a^b\frac{f(x)}{x}dx\approx f(\sqrt{ab})*\ln{\frac{b}{a}}$$ when $a$ and $b$ are close (I don't know how close). Here, close doesn't mean that the differnce between $a$ and $b$ should be 0.01. Here close could mean 'far'. I don't know what the difference should be. Here, close only means that this formula has a limitation. Is it something new? It could help in evaluating $Si(x)$ ( integral of $\frac{\sin{x}}{x}$), $Co(x)$ ( integral of $\frac{\cos{x}}{x}$), and many other integrals whose anti-derivatives are not elementary. I know you can do that by integrating Taylor series of those functions but this seems more compact. Does it work?

Answer 1

With substitution $x=e^t$ and $f(x) = g(t)$, the integral becomes $$ \int_{\ln(a)}^{\ln(b)} g(t)\; dt $$ You can take the Taylor series of $g$ about $t=t_0 = (\ln(a)+\ln(b))/2 = \ln(\sqrt{ab})$: $$ g(t) = g(t_0) + (t-t_0) g'(t_0) + \frac{(t-t_0)^2}{2} g''(t_0) + \frac{(t-t_0)^3}{6} g'''(t_0) + \ldots $$ and integrate it: $$ \int_{\ln(a)}^{\ln(b)} g(t)\; dt = \ln(b/a) g(t_0) + \frac{\ln(b/a)^3}{3} g''(t_0) + \frac{\ln(b/a)^5}{60} g''''(t_0) + \ldots$$