Improper convergent integral with interior singularity For $a$ and $b$ real numbers, $a<b$, I'm looking for an example of a function $f(x)$ that has an infinite discontinuity at some $c$ satisfying $a<c<b$ (i.e. $c$ is in the interior of the domain of integration) and such that
$$\int_a^bf(x)dx$$
is finite. I tried $\int_{-1}^{1}\frac{e^x}{e^x-1}$ but did not work (integral diverges). Thanks.

Both limits, right and left, at $0$ going to $-\infty$:
$$\int_{-1}^{1} \ln (|x|) dx=2$$
One going to $\infty$ and another to $-\infty$:
$$\int_{-1}^{1} \ln (|x|) \text{sgn} (x)dx=0$$
