Compute the value of the integral $\int_1^{\infty} \lfloor x^2 \rfloor e^{-x} \ \mathrm d x $
I provide a somewhat generalised form of the result you have found. Assume that $f(x)$ is integrable on $[1,\infty)$ with antiderivative $F(x)$ and denote $\lim_{x\to\infty}F(x)$ by $F(\infty)$. Then we have \begin{align} &\int_1^\infty\lfloor x^k\rfloor f(x)\,\mathrm{d}x\qquad (k\in\mathbb{N})\\ =&\sum_{n=1}^\infty \int_{n^{1/k}}^\infty f(x)\,\mathrm{d}x\\ =&\sum_{n=1}^\infty \left(F(\infty)-F\left(n^{1/k}\right)\right) \end{align}