This section covers:

Note that **U-Substitution with Definite Integration** can be found here in the **Definite Integration** section.

# Introduction to U-Substitution

**U-Substitution Integration**, or **U-Sub Integration**, is the opposite of the **chain rule**; but it’s a little trickier since you have to set it up like a puzzle. One you get the hang of it, it’s fun, though!

U-sub is only used when the expression with *x* in it that we are integrating isn’t just “*x*”, but is more complicated, like having a coefficient other than 1, such as “4x”. So for example, if a function that we’re raising to an exponent, taking a root of, or taking the trig function of isn’t just “*x*”, we’ll probably have to use u-sub. Here is a function we’d have to use u-sub with: \(\int{{{{{\left( {4x} \right)}}^{5}}}}dx\), because of the “4*x*”.

Why do we have to do something other than just integrate like we learned? Basically, we need U-sub to take the antiderivative of a composite function. Think of the **chain rule**, where we differentiated a **composite function**; for example, for \(\frac{{d\left( {\sin \left( {4x} \right)} \right)}}{{dx}}=4\cos \left( {4x} \right)\), we **multiplied** our “cos(4*x*)” answer by 4, since the derivative of 4*x* is 4. So when we go backward and integrate, it turns out we will have to **divide** by 4. But it’s not always that easy, so we’ll learn some techniques to do the u-substitution.

The reason the technique is called “u-substitution” is because we **substitute** the more complicated expression (like “4*x*” above) with a ** u** (a simple variable), do the integration, and then substitute back the more complicated expression. The “

*u*” can be thought of as the “inside” function. This is also called “change of variables”.

And remember to always **take the derivative back** (if you have time!) to make sure you’ve done the problem correctly. Differentiation tends to be a little easier than integration.

Here is a more “formal” definition:

**U-Substitution Integration Problems**

Let’s do some problems and set up the *u*-sub. The trickiest thing is probably to know what to use as the ** u** (the inside function); this is typically an expression that you are raising to a power, taking a trig function of, and so on, when it’s not just an “

*x*”.

And in the following problems, we are “lucky”; the problems fit into the mold of a “chain rule” problem, so we can easily do the integration. And it’s a good idea to take the derivative back from the answer to make sure we get the integral!

Most *u*-sub problems won’t work exactly like this though; with most *u*-sub problems, we have to somehow get rid of the “extra” variables in the problem by solving for *dx* and canceling them out. If we can do this (sometimes we can’t!), we can solve with *u*-sub. I like to organize the substitutions like this, to really show what’s going on.

Note how we pick the “*u*” to be the expression that is raised to a power, or that we take a root of, or is the argument of a trig function:

One thing that we can notice from the above is a *u*-sub simplification formula we can use, although it’s still good to know the mechanics on how to do the *u*-sub integration. But here goes:

Here are a few more *u*-sub trig problems. Notice in the first problem, we have to separate the \({{\sec }^{4}}x\cdot \tan x\) to \({{\sec }^{3}}x\cdot \sec x\cdot \tan x\) so we can perform the *u*-sub.

Here’s one more problem when we have to solve for both ** dx** in terms of

**and**

*du***in terms of**

*x***to make the**

*u**U*-sub work:

What’s tricky in Calculus around now is that you’ll be expected to figure out **which method of integration** to use; for example, multiplication and separation of terms, *u*-sub, and so on. On a test, they will rarely tell you to us a specific method to integrate, so you’ll have to be able to quickly identify patterns.

So notice how these problems may appear to be *u*-sub, but actually aren’t:

Don’t get discouraged; these take practice!

**Understand these problems, and practice, practice, practice!**

**On to Differential Equations and Slope Fields – you’re ready! **