Problem: Consider the following integral:$I(x) = -\int_{0}^{\infty} \left[ \frac{1}{e^t - 1} - \frac{1}{t} + \frac{1}{2} \right] e^{-xt} \, dt$In the limit$x \to +\infty$, find the asymptotic expansion of the integral up to and including the first three leading orders in z. Provide your answer in a $\boxed{}$ latex environment.
Solution: $\boxed{I(x) \sim -\frac{1}{12x^2} + \frac{1}{120x^4} - \frac{1}{252x^6}}$