Definite integral of a non integrable function(in terms of elementary functions) How can I evaluate something like this:
$$\int _a^{b}\:\frac{1}{\log\left(x\right)}\,dx$$
I've tried introducing a "y" variable to solve it using the differentiation 
technique, but it does not work. So, is there another method to solving this?
Thank you.
 A: Let $I$ be the integral
$$I=\int_a^b\frac{1}{\log(x)}\,dx$$
where it is assumed that the point $x=1$ is not contained in $[a,b]$.
Enforcing the substitution $x= e^{-u}$ yields
$$I=\int_{-\log(a)}^{-\log(b)}\frac{e^{-u}}{u}\,du=-\int_{-\log(b)}^{-\log(a)}\frac{e^{-u}}{u}\,du\tag1$$
The Exponential Integral See Here, $\text{Ei}(x)$, is defined by 
$$\text{Ei}(x)=-\int_{-x}^\infty \frac{e^{-u}}{u}\,du\tag2$$
Using $(2)$ in $(1)$ we see that 
$$I=\text{Ei}(\log(b))-\text{Ei}(\log(a))$$

Hence, the integral $I$ is not expressible in terms of elementary functions, but can be expressed in terms of the special function, the Exponential Integral.

A: The logarithmic or exponential integrals are not an elementary function, but since the logarithm function grows very slowly, the given integral is fairly easy to approximate numerically, especially if the $(a,b)\subset(1,+\infty)$ interval is fairly short. For instance
$$ \int_{2}^{3}\frac{dx}{\log x}\stackrel{\text{IBP}}{=}\left[\frac{x}{\log x}\right]_{2}^{3}+\int_{2}^{3}\frac{dx}{\log^2 x} $$
and $\int_{2}^{3}\frac{dx}{\log^2 x}\approx \frac{1}{\log^2\frac{5}{2}}$. $\text{IBP}$ stands for Integration By Parts, of course.
