Why is this integral (that looks complicated) so easy to evaluate?

Question

I encounter the following integral during Statistics lecture and the professor quickly solved it as if he assumed we know that this integral is easy so solve:

$$ \int_{0}^{\infty} xe^{-(1-t)x} dx = \frac{1}{(1-t)^2}$$

I suspect that this involves a probability distribution but I do not know which. Why is this integral so easy to solve? I know I must have been missing something.

Answer 1

For each $\lambda<0$,\begin{align}\int_0^\infty xe^{\lambda x}\,\mathrm dx&=\left[x\frac{e^{\lambda x}}\lambda\right]_{x=0}^{x=\infty}-\int_0^\infty\frac{e^{\lambda x}}\lambda\,\mathrm dx\\&=-\left[\frac{e^{\lambda x}}{\lambda^2}\right]_{x=0}^{x=\infty}\\&=\frac1{\lambda^2}.\end{align}