Absolute Values and e in an IntegrandDate: 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/ |
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