TITLE: Lesser-known integration tricks
QUESTION [165 upvotes]: I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now looking for a list or reference for some lesser-known tricks or clever substitutions that are useful in integration. For example, I learned of this trick 
$$\int_a^b f(x) \, dx = \int_a^b f(a + b -x) \, dx$$
in the question Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even 
I am especially interested in tricks that can be used without an excessive amount of computation, as I believe (or hope?) that these will be what  is useful for the GRE.

REPLY [6 votes]: I don't know whether these are lesser-known, But here's a list of tricks

For $\int \frac{dx}{ax^2+bx+c}$

if $b^2-4ac \gt0$ use partialfractions. 
if $b^2-4ac =0$ use write it as $(x-\frac{b}{2a})^2$. 
if $b^2-4ac \lt0$ use $tan^{-1}$ by compleating the square.

For $\int \frac{(px+q) \ dx}{ax^2+bx+c}$

if $b^2-4ac \gt0$ use partialfractions. 
if $b^2-4ac \leq0$ make the rearrange numerater in a way it contains the derivative of denominator.

$\int \sin^nx \ dx$ for odd $n$

use change of variable like this $\int \sin^{n-1}x \ \sin xdx=-\int \sin^{n-1}x \ d\cos x$ then since $n-1$ is even you can write it in terms of $\cos x$ using $\sin^2x+\cos^2x=1.$

powers of $tan$,


for odd powers,


$\begin{split} \int \tan^5xdx& = \int \tan^3x\tan^2xdx\\
&=\int \tan^3x(\sec^2x-1)dx\\
&=\int \tan^3x d\tan x-\int\tan^3xdx
\end{split}$ 
use the same method for $tan^3x$ as well.

for even powers,

$\begin{split} \int \tan^2xdx& = \int(\sec^2x-1)dx\\
&=\tan x -x + C
\end{split}$