U-Substitution

U-substitution (also called integration by substitution) is a way to integrate composite functions. It undoes the chain rule. If an integral has the form , set . Then , and

For example, in , the inside function is and its derivative is right there. With , the integral becomes .

Why U-Substitution Works

The chain rule says . Read backwards, as an antiderivative:

Write and . The right side is , which is exactly . So the substitution rule is the chain rule in integral form.

The notation helps too. From , the derivative is , which you can write as . That differential is what replaces in the integral.

How to Choose u

Look for an inside function whose derivative is also a factor of the integrand, up to a constant multiple. Good candidates:

If your first choice doesn’t leave a clean , try another. With practice the right is usually the “innermost” piece of the messiest part.

Steps for U-Substitution

1. Choose , an inside function whose derivative appears as a factor.
2. Find . If needed, solve for the piece of the integrand you’ll replace, such as .
3. Rewrite the whole integral in terms of and . No may be left over.
4. Integrate with respect to .
5. Substitute back and add .
6. Check by differentiating your answer.

Example 1: A Power of an Inside Function

Find .

Let . Then , which is exactly the rest of the integrand:

Check: by the chain rule, . ✓

Example 2: Adjusting for a Constant

Find .

Let . Then , so :

A missing constant is easy to supply this way. A missing variable is not: has no factor, and substitution can’t help.

Example 3: A Linear Inside Function

Find and .

For the first, , :

For the second, , :

That’s the answer the antiderivatives lesson reached by multiplying out. In general, a linear inside function just divides the answer by :

Example 4: A Fraction and the Natural Log

Find .

The numerator is half the derivative of the denominator. Let , so and :

The absolute value can be dropped because . The same idea gives a pattern worth remembering: .

Example 5: The Integral of tan x

Find .

Write and let , so :

Since , this is often written .

Example 6: Use the Indicated Substitution

Many textbooks say “use the indicated substitution to evaluate the integral” and give you .

Evaluate using .

Then :

Example 7: When x Is Left Over

Find .

Let , so . The factor is still there, but you can solve the substitution for it: .

This back-substitution works whenever is easy to solve for , as with any linear .

U-Substitution With Definite Integrals

For a definite integral, there are two correct methods.

Method 1: change the limits. When you change the variable, change the limits too. If , then becomes and becomes :

Then evaluate with the fundamental theorem of calculus in . You never substitute back.

Method 2: substitute back first. Find the antiderivative in as in the indefinite examples, then use the original limits.

Example 8. Evaluate .

Let , . New limits: , and .

Example 9. Evaluate .

Let , , so . New limits: and .

The most common error is mixing the two methods: substituting back to and then plugging in the new -limits, or keeping the old -limits on an integral in .

U-Substitution vs Integration by Parts

Substitution reverses the chain rule, so it fits integrands with a function and its inside derivative. It doesn’t fit a product of two unrelated functions, such as or : no choice of leaves a clean . Those need integration by parts, which reverses the product rule and comes in Calculus 2.

A rule of thumb: if one part of the integrand is the derivative of an inside piece of another part, substitute. If not, think about parts.

Common Mistakes

• Leaving in the integral. After substituting, the integrand must be entirely in . If an remains, solve for or choose a different .
• Forgetting . is , not .
• Dropping the constant factor. If , then . Keep the .
• Trying to fix a missing variable. You can divide by a constant, but not by . can’t be done by substitution.
• Not substituting back in an indefinite integral. The answer must be in terms of .
• Mixing limits in a definite integral. Change the limits, or substitute back, but not half of each.

For more examples, including substitutions with trig and exponential functions, see Section 5.5 of the free textbook OpenStax Calculus Volume 1.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Find .

Problem 2. Find .

Problem 3. Find .

Problem 4. Find .

Problem 5. Find .

Problem 6. Evaluate .

U-substitution finishes the Calculus 1 integration toolkit. Next, the average value of a function uses definite integrals to find the average height of a curve.