How to check if functions are integrable?

Question

Consider two functions

$$ \int_0^1 \frac{1}{e^x-1} dx $$

and

$$ \int_0^1 \frac{1}{(e^x-1)^2} dx $$

How to check if these functions are integrable?

Answer 1

Since the function $$ x \longmapsto\frac{1}{e^x-1} $$ is continuous on $(0,1]$, a potential convergence problem, concerning the integral, is for $x$ near $0$.

We have $$ \frac{1}{e^x-1} \sim_{0^+} \frac1x $$ thus your integral $\displaystyle \int_0^1 \frac{1}{e^x-1} dx$ is divergent as $\displaystyle \int_0^1 \frac{1}{x} dx$ is divergent.

In the same manner, we have $$ \frac{1}{(e^x-1)^2} \sim_{0^+} \frac1{x^2} $$ giving the divergence of $\displaystyle \int_0^1 \frac{1}{(e^x-1)^2} dx$ as $\displaystyle \int_0^1 \frac{1}{x^2} dx$ is divergent.