- #1
JBrandonS
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Homework Statement
##\int_0^\infty \frac{a}{a^2+x^2} dx##
Homework Equations
All the basic integration techniques.
The Attempt at a Solution
So, I saw this problem and wanted to try it using a different method then substitution, which can obviously solve it pretty easy. Since it is a very clear laplace transform I figured I could easily use that but I am getting the wrong answer. Here is what I do:
##
\int_0^\infty \frac{a}{a^2+x^2} dx = \int_0^\infty dx \int_0^\infty e^{-a t}Cos(x t) dt
= \int_0^\infty e^{-a t} \int_0^\infty Cos(x t) dx dt= \int_0^\infty \pi e^{-a t} \delta(t) dt
= \pi (1 - \theta(x))
##The correct answer is:
##\frac{\pi}{2}## iff a > 0
0 iff a = 0
##-\frac{\pi}{2}## iff a < 0
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