"Integration by Substitution" (also called "u-substitution") 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:

Like in this example:

Here

**f=cos**, and we have

**g=x**and its derivative of

^{2}**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:

So

**∫****cos(x**worked out really nicely! (Well, I knew it would.)^{2}) 2x dx = sin(x^{2}) + C
This method only works on some integrals of course, and it may need rearranging:

Now we are ready for a slightly harder example:

And how about this one:

So there you have it.

## In Summary

When we can put an integral in this form:

Then we can make

**u=g(x)**and integrate ∫f(u) du
And finish up by re-inserting

**g(x)**where**u**is.