U-Substitution Integration

This section covers:

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 chain rule; 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 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 “$$4x$$”. 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 “$$4x$$”.

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. Think of the chain rule, where we differentiated a composite function; 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 u-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:

U-Substitution Integration

What this says is that if we want the integral of the outside function, to make it work, we have to make sure that what we’re integrating somehow has a factor that is the derivative of the inside function.

(We can “trick” the integrand into having this factor.)

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 “$$x$$”.

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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!

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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:

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U-Substitution Integration Simplification Formula

$$\displaystyle \int{{{{{\left( {ax+b} \right)}}^{n}}\,dx=\frac{{{{{\left( {ax+b} \right)}}^{{n+1}}}}}{{a\left( {n+1} \right)}}}}+\,C$$

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:

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!

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!

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