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).
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.
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.)
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).
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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[ 3.1 ms ] story [ 34.3 ms ] threadThis 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
I was ticked off about that for some time. (Wouldn't have changed my grade, but it was the principle of the thing.)
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.
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.
http://en.wikipedia.org/wiki/Differentiation_under_the_integ...