Integration by Substitution

"Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") 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 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.

Example: ∫ cos(x 2 ) 2x dx

We know (from above) that it is in the right form to do the substitution:

∫ cos(u) du = sin(u) + C

And finally put u=x 2 back again:

So ∫ cos(x 2 ) 2x dx = sin(x 2 ) + C

That worked out really nicely! (Well, I knew it would.)

Let's just run through that again in a step-by-step manner: