Mathematics

The greatest integer function $[2e^{-x}]$ takes different constant values on different intervals—your job is to find where those breakpoints occur. Since $2e^{-x}$ decreases from $2$ to $0$ as $x$ goes from $0$ to $\infty$, you'll split the integral into regions where $[2e^{-x}]$ equals $0$, $1$, or $2$, then sum the contributions. This transforms a tricky floor-function integral into a sum of simple rectangular areas.

The greatest integer function $[2e^{-x}]$ takes different constant values on different intervals—your job is to find where those breakpoints occur. Since $2e^{-x}$ decreases from $2$ to $0$ as $x$ goes from $0$ to $\infty$, you'll split the integral into regions where $[2e^{-x}]$ equals $0$, $1$, or $2$, then sum the contributions. This transforms a tricky floor-function integral into a sum of simple rectangular areas.