From a Simple Series to Zeta(3): You CAN Learn This Integral

The integral of x²/(e^x + 1) from 0 to ∞ can be evaluated by expanding 1/(e^x + 1) as an alternating geometric series, then integrating term-by-term to obtain the value 3/4 ζ(3), where ζ(3) is Apéry's constant (approximately 1.202). This demonstrates how improper integrals involving exponential functions can be solved through series expansion techniques.