Question

Evaluate $\int {}_0^{\infty }\left[2e^{-x}\right]dx,$ where $\left[x\right]$ is integer function.

Answer 1

Lines $2x+2y=1$

$2x+y=7$

At $(0,\frac{1}{2})$

$L_1:\frac{1}{2}-1=-ve$

$L_2:\frac{1}{2}-7=-ve$

On same side of $(0,\frac{1}{2})$ as they have same signs

The function $y=2e^x$ is a monotonically decreasing function

At $x=0$ we have $y=2$

At $x=\ln 2$, $y=1$

Hence for all $x>\ln 2$, $y<1\Rightarrow \left[y\right]=0$

So the integral

$I={\int }_0^{\infty }\left[y\right]dx={\int }_0^{\ln 2}1dx$