All right, you've learned all the basic integral calculus formulas, and you're ready to take on some problems! Well, not quite. You've still got to get at least one more skill under your belt: u-substitution. However, don't let this become a bad thing in your mind! It's quite nice to see complex integrals suddenly become basic.

Before you read on, please make sure you know the basic integration rules. The focus of this article will be on indefinite integrals only, but solving definite integrals with u-substitution uses the same steps. In addition, if you are looking for another way to solve integrals, please read my integration by parts article.


How u-substitution works
Like any form of mathematical substitution you've seen before, there's nothing magical about u-substitution for calculus integrals. You are simply going to simplify the indefinite integral by...

  1. Replacing a portion of the integrand with u so that
  2. Its derivative encompasses the rest of the integrand, and
  3. Using integration rules, solve the integral, then finally
  4. Revert u back to being in terms of x

Our example problem
The first step to solving calculus integrals is to, first, write down the integral. Here it is:
The problem
Our goal is to solve this indefinite integral using u-substitution. Let's get started by taking the 9 out to avoid getting it in the way.

Then let's start substituting!


1. Replace a portion of the integrand with u
We essentially have three choices for u. We could replace csc x, csc2x, or cot x with u; however, only one of these will lead you to the answer. We use step 2 to determine which one it is.


2. So that its derivative encompasses the rest of the integrand
The question is simply, "Whose derivative will get the entire integrand in terms of u?" The answer is cot x's, whose derivative is csc2x. After getting everything in terms of u and du, there is still a bit of rearranging that has to be done. See the steps below:

Using u-substitution


3. Using integration rules, solve the integral
This integral should be recognizable to you as it is the derivative of ln |u|. Therefore,

Solving the calculus integral


4. Revert u back to being in terms of x
Now that you've solved the integral, all that is left for you to do is replace u with cot x.

Writing u in terms of x again


Conclusion
That's all! u-Substitution is a neat little integral calculus trick that works for most of the calculus problems that are designed in textbooks. The hardest part is finding which u is the correct one to choose. So, in summary, here are some pointers:
  • du must be in the numerator.
  • Everything must be in terms of u.
  • Remember to revert u back into terms of x before you finish the problem!
And with that, I say, good luck with your studies!