Killer Integrals: some clever integrals 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. 


*

*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.

*The scarier looking, generally, the better.

*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? 
 A: 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:
$$\int_2^4\!\!\frac{\sqrt{\ln(9 - x)}}{\sqrt{\ln(9 - x)} + \sqrt{\ln(x + 3)}}\,\mathrm{d}x $$ 
A: Though you may already know this (and this is more of a trick rather than a specific integral), one of my favorite tricks is symmetric integration. All this requires is knowledge of even and odd functions.
Remember: An even function is symmetric about the y-axis; An odd function is symmetric about the origin (or the line $y=x$, whatever you prefer). 
If $f$ is even, then
$$\int_{-a}^{a}f(x) \text{ d}x = 2\int_{0}^{a}f(x) \text{ d}x$$
If $f$ is odd, then 
$$\int_{-a}^{a}f(x) \text{ d}x = 0$$
This could easily be visualized by graphing an even function and an odd function and shading in the area over an interval, [-a, a].
Also, Weierstrass Substitution is a great method for solving those crazy yet beautiful integrals.
Basically, by substituting $\tan(t/2)=x$, we get
$$\sin(x)=\frac{2t}{1+t^2}$$
$$\cos(x)=\frac{1-t^2}{1+t^2}$$
$$\text{d}x=\frac{2}{1+t^2}$$
Again, I know these are all methods and the question asked for specific integrals, BUT these are great (and easy to visualize/derive) ways to solve those integrals.
