I've finished Calculus 1 and I'm going to go into Calculus II next semester, so I've been trying to keep my integration skills up to par. I noticed that in some parts of integration in Calculus 1, there are some integrals that, with a clever $u$ substitution turn into something beautiful.
I know beautiful is a very subjective term, and so is clever, but given the nature of this question, I would like to give a couple of pointers as to what I'm looking for.
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It has got to have some kind of a clever trick, this could be in the form of something as simple as understanding a visualization that's going on, or a very clever identity.
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The scarier looking, generally, the better.
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The solution or process that leads to the solution has some slick clever trick that is guaranteed to put a smile on even the most bitter mathematician's face.
As an example, when I first started integrating, I learned that:
$$\int \frac{1}{x^2+1} = \tan^{-1}(x)$$
And I thought that was beautiful, because I wouldn't have expected that answer. Then I learned that when you have some integral of the form of :
$$\int \frac{1}{x^2+49} = \frac{1}{7}\tan^{-1}\left(\frac{x}{7}\right)$$
And then I tried more patterns following the same something squared term and it follows a beautiful pattern. But none of that was expected, at first.
Make sense?
Best Answer
The following integral is nice - first you need to prove convergence, and then you evaluate it through a nice substitution:
$$\int_{0}^{\pi/2}\!\!\ln(\sin{x})\,\,\mathrm{d}x $$
Another similar integral is $$\int \!\sqrt{\tan{x}}\,\,\mathrm{d}x$$
Oh, and here's a really easy integral (from the $1987$ Putnam) that looks rather intimidating: $$\int_2^4\!\!\frac{\sqrt{\ln(9 - x)}}{\sqrt{\ln(9 - x)} + \sqrt{\ln(x + 3)}}\,\mathrm{d}x $$