U-Substitution Integration




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 “4x”.

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(4x)” answer by 4, since the derivative of 4x 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 “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 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!

Simple U-Sub Problems

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:

u-sub-integration-problems

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:

U-sub Simplification

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.

u-sub-trig-problems

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:

u-sub-solving-for-x

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:

No Usub

Don’t get discouraged; these take practice!


Understand these problems, and practice, practice, practice!


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