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Re: Incorrect symbolic improper integral

• To: mathgroup at smc.vnet.net
• Subject: [mg103606] Re: [mg103586] Incorrect symbolic improper integral
• From: Bayard Webb <bayard.webb at mac.com>
• Date: Wed, 30 Sep 2009 04:59:25 -0400 (EDT)
• References: <200909291138.HAA25632@smc.vnet.net>

I think you need to add a as a coefficient of x everywhere, including  
the squared term.

In[6]:= Assuming[a \[Element] Reals,
  Integrate[Cos[a x]/(1 + (a x)^2), {x, -\[Infinity], \[Infinity]}]]

Out[6]= \[Pi]/(E Abs[a])

Setting a = 1 yields Mathematica's previous result.

Bayard

On Sep 29, 2009, at 4:38 AM, jwmerrill at gmail.com wrote:

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: Re: Incorrect symbolic improper integral
    • From: "King, Peter R" <peter.king@imperial.ac.uk>

• References:
  • Incorrect symbolic improper integral
    • From: "jwmerrill@gmail.com" <jwmerrill@gmail.com>