5 ms·

Integration by integration under the integral sign

52 points by nilaykumar 15y ago

Bo102010 15y ago
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 http://math2.org/math/integrals/more/sec.htm
- nilaykumar 15y ago
  Haha, yeah. That one's pretty cool too. There are a couple of slightly more easier-to-see ways, but they still involve seemingly arbitrary multiplications of cosine, etc (or product-to-sum rules). Man... integrals. Love/hate relationship.
- btilly 15y ago
  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.)
mturmon 15y ago
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 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.
- nilaykumar 15y ago
  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).
trentonstrong 15y ago
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.
- acqq 15y ago
  Thanks, here's the full Feynman's reference at the end of this article: http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign http://en.wikipedia.org/wiki/Differentiation_under_the_integ...
- nilaykumar 15y ago
  This is going on my books to check out as _soon_ as I get to college list!

52 points by nilaykumar 15y ago

Bo102010 15y ago

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 http://math2.org/math/integrals/more/sec.htm

nilaykumar 15y ago
Haha, yeah. That one's pretty cool too. There are a couple of slightly more easier-to-see ways, but they still involve seemingly arbitrary multiplications of cosine, etc (or product-to-sum rules). Man... integrals. Love/hate relationship.
btilly 15y ago
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.)

mturmon 15y ago

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 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.

nilaykumar 15y ago
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).

trentonstrong 15y ago

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.

acqq 15y ago
Thanks, here's the full Feynman's reference at the end of this article: http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign http://en.wikipedia.org/wiki/Differentiation_under_the_integ...
nilaykumar 15y ago
This is going on my books to check out as _soon_ as I get to college list!