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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.

  1. 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.

  2. The scarier looking, generally, the better.

  3. 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 :

$$\int \frac{1}{x^2+49} = \frac{1}{7}\cdot\tan^-1\left(\frac{x}{7}\right)$$

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?

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    $\begingroup$ integral-table.com Go crazy. $\endgroup$ – Shahar Mar 29 '14 at 21:33
  • $\begingroup$ $\int \frac{dx}{x^2+a^2} = 1/a \arctan (x/a)$ $\endgroup$ – Ganesh Mar 29 '14 at 21:34
  • $\begingroup$ @Ganesh I noticed that. $\endgroup$ – alvonellos Mar 29 '14 at 21:41
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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: $$\int_2^4\!\!\frac{\sqrt{\ln(9 - x)}}{\sqrt{\ln(9 - x)} + \sqrt{\ln(x + 3)}}\,\mathrm{d}x $$

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