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Incorrect symbolic improper integral

• To: mathgroup at smc.vnet.net
• Subject: [mg103586] Incorrect symbolic improper integral
• From: "jwmerrill at gmail.com" <jwmerrill at gmail.com>
• Date: Tue, 29 Sep 2009 07:38:28 -0400 (EDT)

Below is a definite integral that Mathematica does incorrectly.
Thought someone might like to know:

In[62]:= Integrate[Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}]

Out[62]= \[Pi]/E

What a pretty result--if it were true. The correct answer is \[Pi]*Cosh
[1], which can be checked by adding a new parameter inside the
argument of Cos and setting it to 1 at the end:

In[61]:= Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \[Infinity]},
  Assumptions -> a \[Element] Reals]

Out[61]= \[Pi] Cosh[a]

Regards,

Jason Merrill

• Follow-Ups:
  • Re: Incorrect symbolic improper integral
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