Introduction to U-Substitution | Which Method to Use? |

U-Substitution Integration Problems | More Practice |

Note that **U-Substitution with Definite Integration** can be found here in the **Definite Integration** section, **U-Substitution** with **Exponential and Logarithmic Integration** can be found in the **Exponential and Logarithmic Integration** section, and **U-Substitution with Inverse Trig Functions** can be found in the **Derivatives and Integrals of Inverse Trig Functions** section.

## Introduction to U-Substitution

**U-Substitution Integration**, or **U-Sub Integration**, is the opposite of the **The Chain Rule** from **Differential Calculus**, but it’s a little trickier since you have to set it up like a puzzle. Once you get the hang of it, it’s fun, though! U-sub is also known the **reverse chain rule** or **change of variables**.

Why do we have to do something other than just integrate like we learned? Basically, we need **U-sub** to take the anti-derivative of a **Composite Function**; it’s the “undoing” of the **Chain Rule**. For example, for $ \displaystyle \frac{{d\left( {\sin \left( {4x} \right)} \right)}}{{dx}}=4\cos \left( {4x} \right)$, we **multiplied** our “$ \cos \left( {4x} \right)$” answer by **4**, since the derivative of $ 4x$ is **4**. 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 substitution.

The reason the technique is called “**U-substitution**” is because we **substitute** the more complicated expression (like “$ 4x$” 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; don’t worry if you don’t get this; you won’t 🙂 :

**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$”. Sometimes, but not always, it’s the function inside parentheses or under a radical sign.

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. Again, 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 start by picking the “$ u$” to be the expression that is raised to a power, or that we take a root of, or is the argument of a trig function (for example, “$ 4x$”):

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 $ du$ and $ x$ in terms of $ u$ to make the $ u$-sub work. Try doing something like this when you otherwise get stuck:

## Which Method to Use?

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.

The following problems appear to be $ u$-sub, but actually aren’t. I like to try other methods first before trying $ u$-sub:

Don’t get discouraged; these take practice!

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

**For Practice**: Use the **Mathway** widget below to try a **U-Sub** problem. Click on **Submit** (the blue arrow to the right of the problem) and click on **Describe the Transformation** to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on **Tap to view steps**, or **Click Here**, you can register at **Mathway** for a **free trial**, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

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