

Integration by integration under the integral sign - nilaykumar
http://1over137.wordpress.com/2011/06/04/integration-by-integration-under-the-integral-sign/

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mturmon
I think the standard derivation of the Gaussian integral uses the same
technique. It's between equations (2) and (3) of

<http://mathworld.wolfram.com/GaussianIntegral.html>

You make the single integral into a double integral, and then the double
integral turns out to be easy to do using polar coordinates.

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nilaykumar
Oh, that's a very good point! I never really thought of that technique as this
kinda trick, but yeah. For some reason I always thought of it as squaring the
integral (or the contour integration method).

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Bo102010
This was pretty cool. Once on an exam in a differential equations class, the
instructor made a mistake in the problem that required integrating sec(x).

This is a fairly difficult integral, since it requires the trick of
multiplying by 1 in a clever way.
<http://math2.org/math/integrals/more/sec.htm>

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btilly
In my first Calculus course I had failed to memorize some trig integral, and
rederived it on the spot. Unfortunately what I came up with differed from the
usual by some complex trigonometric substitution. The grader saw a ton of
work, not the usual answer, and didn't bother differentiating it to
demonstrate that it was correct, and marked me wrong.

I was ticked off about that for some time. (Wouldn't have changed my grade,
but it was the principle of the thing.)

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trentonstrong
Wow, thanks for posting this.

I was equally inspired during undergrad and grad school by Feynman's "unusual
tools", and ended up checking out a copy of Advanced Calculus by Woods which
was apparently the book he used to learn calculus in high school. If I recall
correctly, Woods goes over integration by both differentiation and integration
under the integral sign, including some interesting ways to set boundary
conditions to get the answer you want. It's a nice book if you ever get the
chance.

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acqq
Thanks, here's the full Feynman's reference at the end of this article:

[http://en.wikipedia.org/wiki/Differentiation_under_the_integ...](http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign)

