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

Answer 1

Ezra Adams

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#

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)#

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Answer 2

Christopher Agresta

To evaluate the integral ∫e^(-|x|) from -∞ to ∞, you can split the integral into two parts: from -∞ to 0 and from 0 to ∞. In each part, the absolute value function can be removed by considering the different cases of x being negative or positive. After integrating each part separately, you will arrive at the result of 2.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.