Question

Evaluate the integrals :
$\underset{0}{\overset{\infty }{\int }}\left[\frac{2}{e^x}\right]dx$, where [.] denotes the greatest integer function.

Answer 1

Let $y=\underset{0}{\overset{\infty }{\int }}\left[\frac{2}{e^x}\right]dx$

Then, $y=\underset{0}{\overset{\infty }{\int }}\left[2e^{-x}\right]dx$

At $x=0,$ $y=2e^0=2$

At $x=ln\left(2\right),$ $y=1$

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

So the integral

$I=\underset{0}{\overset{\infty }{\int }}\left[y\right]dx$

$I=[x{]}_0^{ln\left(2\right)}$

$I=\underset{0}{\overset{ln\left(2\right)}{\int }}1dx$

$I=ln\left(2\right)-0$

$I=ln\left(2\right)$