Absolute Values and e in an Integrand

Date: 05/10/98 at 10:06:57
From: Raymond
Subject: Integral of e^(-|x|)

I have recently returned to school to get a masters degree in software 
engineering. It has been a while since I have had to use the 
techniques of integration that I learned in my undergraduate years. 
Would you please tell me which integration technique I must use to 
find the integral of e^(-|x|) dx?

Date: 05/10/98 at 11:09:30
From: Doctor Sam
Subject: Re: Integral of e^(-|x|)

Raymond,

In general, functions involving absolute values are best broken up 
into two different functions and then worked through as two separate 
cases. In this problem, if x > 0, then |x| can just be replaced by x 
and you are looking for INT(e^-x) dx.

When x < 0, |x| can be replaced by -x (the opposite of x). In this 
case, you are looking for INT(e^x) dx.

This last integral is a standard form. The derivative of (e^x) = e^x, 
so INT(e^x) dx = e^x + C.

The first integral, INT(e^-x) dx, equals -e^(-x) + C, which you can 
reason out from the derivative formula for [e^(-x)]' = - e^(-x).

Therefore:

   when x > 0, INT(e^(-|x|)) dx = -e^(-x) + C 

and

   when x < 0, INT(e^(-|x|)) dx = e^(x) + C

When you are trying to break down a function involving absolute 
values, ask yourself, "When does the quantity inside the absolute 
value change signs?" That way, if you are faced with a problem like 
INT(e^|2x-5|) dx, you can determine that x = 2.5 is the critical 
value. When x > 2.5, the quantity |2x - 5| = 2x - 5, since positive 
numbers are the same as their absolute values. But when x < 2.5, 
|2x - 5| = -(2x -5) = -2x + 5, since negative numbers are the opposite 
of their absolute values.

I hope that helps.

-Doctor Sam, The Math Forum
Check out our web site! http://mathforum.org/dr.math/