I am working on a problem in my past Qual.
"Prove that $e^{-x}$ is Lebesgue integrable on $[0,\infty)$ and compute the integral."
Here is my solution: $e^{-x}$ is continuous, hence measurable. We can use the monotone convergence theorem $$\int_0^\infty e^{-x} = \int_0^n \lim(e^{-x}\chi_{[0,n]})=\lim \int_0^n(e^{-x})=1<\infty$$ So it is Lebesgue integrable.
This solution seems too short and direct. So I doubt that it is wrong.