Integral in a variational principle problem

[MrMuscle]

TL;DR

Trying to solve the integral, for variational principle in griffith's book.
But I'm stuck at the final step. Can you please help?

Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't go further. A little bit help to finish the integration would do great! Thanks for your help in advance!

 

[PeterDonis]

@MrMuscle please post your own equations directly in the thread using the PF LaTeX feature, not in an image. We cannot respond to equations that are in an image.

Help on the PF LaTeX feature can be found here:

https://www.physicsforums.com/help/latexhelp/

 

[tnich]

Summary: Trying to solve the integral, for variational principle in griffith's book.
But I'm stuck at the final step. Can you please help?

Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't go further. A little bit help to finish the integration would do great! Thanks for your help in advance!
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You could make $e^{-2bx^2}$ look like a Gaussian zero-mean probability density function and use known results for the area under the curve and the variance.

 

[PeroK]

Summary: Trying to solve the integral, for variational principle in griffith's book.
But I'm stuck at the final step. Can you please help?

Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't go further. A little bit help to finish the integration would do great! Thanks for your help in advance!
View attachment 253010
View attachment 253009

If you look inside the back cover of the Griffiths book you might find those standard integrals.

If you want to derive them yourself, the first can be done by a clever trick and transforming to polar coordinates; and the second can be reduced to the first using integration by parts.

PS I'm not sure how you got so far in the book without doing those integrals about 20 times already!