TITLE: What are your favorite integration tricks?
QUESTION [5 upvotes]: 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:
$$\int_{-a}^{a}{f(x)}dx=0 \space\text{if}\space f(x) \space\text{is odd}$$

REPLY [2 votes]: 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 :
$$\begin{align} \int_0^{\pi} \frac{x\sin(x)}{1+\cos^2 (x)}\mathrm{d}x &=\frac{\pi}{2}\int_0^{\pi} \frac{\sin(x)}{1+\cos^2 (x)}\mathrm{d}x\\ &=\frac{\pi}{2} \left[-\arctan(\cos(x))\right]_0^{\pi} \\ &=\frac{\pi^2}{4}\end{align}$$