Help me to find a Definite Integral

[sunveer]

1. Evaluate [PLAIN]http://www.goiit.com/equations/2011/5/25/795493e8-6a4d-42b0-8c18-2afa4e75b653.png[/b]



I have tried hard to solve it but I am not getting the exact answer.
The ans is given to be -a^2/(1+a^2)

 

[HallsofIvy]

Integration by parts. Let u= e^{-2x}[/tex] and dv= cos(2ax)<br /> <br /> The result will involve <br /> \int e^{-2x}sin(2ax)dx&amp;lt;br /&amp;gt; so use integration by parts again. Let u= e^{-2x} again but let dv= sin(2ax).&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; That will give you a formula that again involves&amp;amp;lt;br /&amp;amp;gt; \int e^{-2x}cos(2ax)dx &amp;amp;lt;br /&amp;amp;gt; so set the whole thing equal to that integral.&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; Try that and if you have a problem show your work here.

 

[sunveer]

I solved it by Integration by parts
but
at last I am getting

(1+a^2){∫e−2xcos(2ax)dx}=0

 

[Ray Vickson]

1. Evaluate [PLAIN]http://www.goiit.com/equations/2011/5/25/795493e8-6a4d-42b0-8c18-2afa4e75b653.png[/b]



I have tried hard to solve it but I am not getting the exact answer.
The ans is given to be -a^2/(1+a^2)

Your integral is divergent. However, the integral with exp(-2*|x|) instead of exp(-2*x) does give 1/(1+a^2), and the integral with exp(-x^2) instead of exp(-2*x) gives sqrt(pi) * exp(-a^2).

RGV