"Integration by Substitution" (also called "usubstitution") is a method to find an integral, but only when it can be set up in a special way.
The first and most vital step is to be able to write our integral in this form:
Note that we have g(x) and its derivative g'(x)
Like in this example:
Here f=cos, and we have g=x^{2} and its derivative of 2x
This integral is good to go!
When our integral is set up like that, we can do this substitution:
Then we can integrate f(u), and finish by putting g(x) back as u.
Like this:
We know (from above) that it is in the right form to do the substitution:
Now integrate:

And finally put u=x^{2} back again:
sin(x^{2}) + C
worked out really nicely! (Well, I knew it would.)
This method only works on some integrals of course, and it may need rearranging:
Example:
Oh no! It is 6x, not 2x. Our perfect setup is gone.
Never fear! Just rearrange the integral like this:
(We can pull constant multipliers outside the integration, see Rules of Integration.)
Then go ahead as before:
Now put u=x^{2} back again:
3 sin(x^{2}) + C
Done!
Now we are ready for a slightly harder example:
Example:
Let me see ... the derivative of x^{2}+1 is 2x ... so how about we rearrange it like this:
Then we have:
Then integrate:
Now put u=x^{2}+1 back again:
Â½ ln(x^{2}+1) + C
And how about this one:
Example:
Let me see ... the derivative of x+1 is ... well it is simply 1.
So we can have this:
Then we have:
Then integrate:
Now put u=x+1 back again: