Smart Integration Tricks I am in the last year of my school 
and studied integration this year 
I have done several Integration techniques like Integration


*

*By substitution

*By partial fractions

*By parts

*Trigo. substitutions

*Tangent half angle substitution
and many other basic methods of integration.


So I wanted to ask about some integration tricks that might prove quite helpful.
Not something advanced which is taught at higher level of studies 
But some smart integration tricks at school level only.

One very simple example that pops to mind (several of my students have been dumbfounded on first sight with the following example).
Find the anti derivative (there are many such examples)
$$\int \frac{t}{t+1}dt $$
You add and subtract $1$ in the numerator and voila.

Another one. Bearing in mind $\int \frac{1}{1+t^2}dt = \arctan t + C$. Find the anti derivative, say,
$$\int \frac{1}{9t^2 -6t +2}dt $$
Separate the square part and integrate with respect to $3t-1$. Simplifies to finding
$$\frac{1}{3}\int \frac{1}{(3t-1)^2 +1}d(3t-1)  $$
