Integration Using u-Substitution: An Introduction

What u-Substitution Is For

Some integrals are easy because they match a basic antiderivative rule directly. For example, is straightforward because it matches the reverse power rule.

But many integrals contain a function inside another function. For example, consider . The expression is nested inside the power. The reverse power rule by itself does not immediately handle this structure.

u-substitution is a method for simplifying these integrals by replacing a complicated inside expression with a single variable, usually . It is essentially the integration version of the chain rule.

The Basic Idea

Suppose an integral contains a function together with its derivative . We can make the substitution:

Then differentiate both sides:

The goal is to replace both the complicated expression and its differential with simpler expressions involving . Once the substitution is made, the integral often becomes a basic integral that you already know how to evaluate.

The u-Substitution Formula

The general pattern is:

When you see an integral that resembles this pattern, look for an inside function whose derivative also appears somewhere in the integrand.

A Step-by-Step Method

1. Choose . Look for the complicated inside expression. This is often the expression raised to a power, inside a square root, inside an exponential, or inside a trigonometric function.
2. Find . Differentiate your chosen to obtain .
3. Rewrite the integral in terms of . Replace the appropriate part of the integrand with .
4. Integrate with respect to . The difficult integral should now be simpler.
5. Substitute back. Replace with the original expression.
6. Add . Because this is an indefinite integral, don't forget the constant of integration.

Worked Examples

How to Recognize a u-Substitution Problem

The hardest part of u-substitution is often not the integration itself. It is recognizing when substitution will work.

Look for a composition: an expression inside another function. Then ask whether the derivative of that inside expression also appears in the integrand, possibly multiplied by a constant.

For example, in , the inside function is . Its derivative is , and the integrand contains exactly that factor. That is a strong signal that u-substitution is appropriate.

A useful mental question is:

When a Constant Is Missing

Sometimes the derivative of your proposed is present only up to a constant multiple.

For example:

If we let , then . The integrand contains , which is . Therefore, we can account for the extra factor by multiplying by .

This is one of the most important practical skills in u-substitution: the derivative does not have to appear exactly as ; it may differ by a constant factor.

Definite Integrals and u-Substitution

u-substitution also works for definite integrals. There are two common approaches: substitute back to before applying the limits, or change the limits from to .

Changing the limits can make the work cleaner.

Common Mistakes

Practice Problems