Sure! Here's an extremely difficult and short calculus problem for you:

Problem: Evaluate the following integral:

$\int_0^\infty \frac{x^2}{e^x - 1} \, dx$

This is a challenging integral that involves advanced techniques such as the use of the Gamma function and Zeta function.

Let me know if you'd like a detailed explanation or if you have any questions about it!

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Here are 5 related questions to explore the topic further:

1. How does the Gamma function relate to integrals of this form?
2. What is the derivation of the integral form of the Riemann Zeta function?
3. How do we handle integrals with singularities at infinity in calculus?
4. What role do series expansions play in solving integrals involving exponential functions?
5. Can you generalize this integral to other powers of $x$?

Tip: Remember to look into Laplace transforms when dealing with integrals of exponential functions.