In specific, I'm supposed to decide whether the following integral exists, and if it does, to calculate its value:
$$\int_0^\infty \frac{x\ln(x)}{\sinh(x)-x}dx.$$
After checking with Mathematica, I came to the conclusion that this integral does not exist. Furthermore, it appears that the integrand does not possess an elementary primitive.
Up until now, I only came across divergent integrals where one could calculate the primitive, and upon evaluating at the boundaries and taking the limit from t to c or infinity, one finds that the integral diverges.
The integral converges "at infinity" but diverges "at zero".
As $x\to 0$, $\sinh x-x\sim x^3/6$ so $$\frac{x\ln x}{\sinh x-x}\sim\frac{6\ln x}{x^2}$$ and the improper integral $$\int_0^1\frac{\ln x}{x^2}\,dx$$ diverges.