Question

How do you evaluate the integral `int e^(-absx)` from `-oo` to `oo`?

Answer 1

2

Explanation:

graph{e^(- | x|) [-10, 10, -5, 5]}

use the symmetry so that it becomes

`color{red}{2 times} int_0^oo dx qquad e^{-x}`

ie we are integrating in the region `x>=0` using the fact that `|x| = x`

`= 2 [- e^{-x}]_0^oo`

`= 2 [ e^{-x}]_oo^0`

2

to test for the symmetry use the even funcition test ie does `f(-x) = f(x)`

here

`f(-x) = e^{- abs (-x)} = e^{- abs x}= f(x)`