A solution Integral of the Week #2 [Walking Randomly]

I’ve just submitted the following solution for the

« Integral of the Week #2

The second Integral Of The Week (IOTW) is rather different from the first in that I am going to give you the evaluation. Your task is to prove it.

\displaystyle\int_{-\infty}^{\infty}e^{-x^2}\; dx=\sqrt{\pi}

But WAIT! Almost every time I have seen this integral evaluated, it has been done by squaring it and converting to polar co-ordinates and that’s the one method of evaluation you can’t use for this particular challenge. I am looking for more ‘interesting’ proofs. Have fun.

Solutions can be posted in the comments section or sent to me by email (obtaining my email address is another puzzle for you to solve) and will be discussed in a future post. Feel free to send your solution in just about any format you like – plain text, uncompiled Latex, PDF, postscript, Mathematica, ODF, even Microsoft Word. When I get around to posting the solutions I will attempt to standardize them (to PDF probably).

By the way – you still have time to submit a solution for the first IOTW.

=\displaystyle\int_{0}^{\infty }\dfrac{-e^{-\underset{x\rightarrow \infty}{\lim }x}+e^{0}}{2\left( 1+y^{2}\right) }\; dy=\displaystyle\int_{0}^{\infty }\dfrac{1}{2\left( 1+y^{2}\right) }\; dy
=\dfrac{1}{2}\left[ \tan^{-1}y\right] _{y=0}^{\infty }=\dfrac{1}{2}\left( \dfrac{\pi}{2}-0\right) =\dfrac{\pi}{4}.

[updated at 14:41 GMT: two integrals corrected]

[update of July 26, 2008: \square added]

Also posted in my main blog problemas | teoremas

A solution Integral of the Week #2 [Walking Randomly]

Deixe uma Resposta

Preencha os seus detalhes abaixo ou clique num ícone para iniciar sessão:

Logótipo da WordPress.com

Está a comentar usando a sua conta WordPress.com Terminar Sessão / Alterar )

Imagem do Twitter

Está a comentar usando a sua conta Twitter Terminar Sessão / Alterar )

Facebook photo

Está a comentar usando a sua conta Facebook Terminar Sessão / Alterar )

Google+ photo

Está a comentar usando a sua conta Google+ Terminar Sessão / Alterar )

Connecting to %s