Question

${\int }_0^{\infty }\left[e^{|-x|}\right]dx$ is equal to (where [.] denotes the greatest integer function)

Answer 1

We have,

$I={\int }_0^{\infty }{e}^{|-x|}dx$

We know that,

$x\ge 0\left|x\right|=x$

$x\le 0\left|x\right|=-x$

Thus,

$I={\int }_0^{\infty }{e}^{|-x|}dx$

$={\int }_0^{\infty }{e}^{x}dx+{\int }_0^{\infty }{e}^{-x}dx$

$={{\left[e^x\right]}_0}^{\infty }+{{\left[e^{-x}\right]}_0}^{\infty }$

$=e^{\infty }-e^0+e^{-\infty }-e^{-0}$

$=0-1+0-1$

$=-2$

Hence, this is the answer.