What are your favorite integration tricks? I'm learning to integrate and I'd like to hear what are you favorite integration tricks?
I can't contribute much to this thread, but I like the fact that:
$$\int_{-a}^{a}{f(x)}dx=0 \space\text{if}\space f(x) \space\text{is odd}$$
 A: Integration by parts.
There is even a separate thread on stack-exchange on integration by parts.
Striking applications of integration by parts
The discrete analog of integration by parts i.e. the summation by parts is also an important tool especially in analytical number theory when we want to find asymptotic. For instance, my recent post here, uses this to get an estimate of $\displaystyle \sum_{n=N+1}^{\infty} \dfrac1{n^s}$.
A: *

*Residues.  They're fantastic. One small part of this is the evaluation of a rational function of $\sin(x), \cos(x)$. By making a simple substitution, you can reduce such integrals to 'ordinary' ones, and then use residues. 

A: One I really like is this one : 
If $f$ is a continuous function for which $f(a+b-t)=f(t)$ then $$\int_a^b t\cdot f(t) \mathrm{d}t=\frac{a+b}{2}\int_a^bf(t) \mathrm{d}t$$
Example :
$$\begin{align} \int_0^{\pi} \frac{x\sin(x)}{1+\cos^2 (x)}\mathrm{d}x &=\frac{\pi}{2}\int_0^{\pi} \frac{\sin(x)}{1+\cos^2 (x)}\mathrm{d}x\\ &=\frac{\pi}{2} \left[-\arctan(\cos(x))\right]_0^{\pi} \\ &=\frac{\pi^2}{4}\end{align}$$
