Integration by Substitution
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=x2 and it's derivative 2xThis 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.We know (from above) that it is in the right form to do the substitution:
Now integrate:∫cos(u) du = sin(u) + CAnd finally put u=x2 back again:sin(x2) + C (Math is Fun, n.d)
Note that we have g(x) and its derivative g'(x).
Like in this example:
Here f=cos, and we have g=x2 and it's derivative 2xThis 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.We know (from above) that it is in the right form to do the substitution:
Now integrate:∫cos(u) du = sin(u) + CAnd finally put u=x2 back again:sin(x2) + C (Math is Fun, n.d)