u-Substitution with Exponential and Logarithmic Functions

What u-Substitution Is For

u-substitution is one of the most important integration techniques in AP Calculus. It is essentially the reverse of the chain rule. Whenever an integral contains a function together with its derivative, u-substitution can often simplify the problem dramatically.

This is especially useful with exponential and logarithmic functions. Expressions such as , , and may look complicated at first, but a well-chosen substitution can reduce them to standard integrals that you already know.

The basic idea is simple: choose the complicated inner expression, call it , and rewrite the integral in terms of .

The Basic Formula

Suppose you have an integral of the form:

If you let , then . The substitution transforms the original integral into:

The goal is to replace the complicated expression with a single variable so that the integral becomes easier to recognize and evaluate.

Why Exponential Functions Work So Well

The function has a special property: its derivative is itself. This makes exponential integrals particularly friendly to u-substitution.

For example, consider an integral containing . The expression is the natural choice for because its derivative is simply .

After substitution, the complicated exponential expression disappears and you are left with the basic integral of .

Why Logarithmic Functions Work So Well

The natural logarithm also appears frequently in u-substitution problems. Remember that:

This means that an integral containing a complicated expression in a denominator may become a logarithm after substitution.

A particularly important pattern is:

Whenever you see an expression of the form multiplied by something resembling , you should immediately consider u-substitution.

A Step-by-Step Method

Choose u. Look for an inner expression that is being used inside an exponential, logarithm, power, or denominator.
Differentiate u. Find .
Rewrite the integral. Replace the original expression with and .
Integrate with respect to u. The resulting integral should be simpler.
Substitute back. Replace with the original expression.
Check by differentiating. This is especially useful when the integral contains several layers of composition.

Worked Examples

Example 1: Exponential Function

Evaluate .

Step 1: Choose the inside expression:

Step 2: Differentiate:

Step 3: Rewrite the integral:

Step 4: Integrate:

Result:

Example 2: Exponential Function with a Coefficient

Evaluate .

Step 1: Let .

Step 2: Then , so .

Step 3: Substitute:

Step 4: Integrate:

Result:

Example 3: Natural Logarithm

Evaluate .

Step 1: Choose:

Step 2: Differentiate:

Step 3: Substitute:

Step 4: Integrate:

Result:

Example 4: Logarithm with a Polynomial Inside

Evaluate .

Step 1: Let .

Step 2: Then .

Step 3: Rewrite:

Step 4: Integrate:

Result:

Example 5: Exponential Function with a Nonlinear Exponent

Evaluate .

Step 1: Let .

Step 2: Then , so .

Step 3: Substitute:

Step 4: Integrate:

Result:

Recognizing the Pattern

The most important skill is not performing the algebra. It is recognizing when u-substitution is appropriate.

Look for these common patterns:

In each case, the key question is the same: Can I identify an inner expression whose derivative also appears somewhere in the integral?

What If the Derivative Is Almost There?

Sometimes the derivative of your chosen is present except for a constant factor. This is not a problem. You can compensate by multiplying by the appropriate constant.

For example, consider:

If , then . The integral contains , which is twice as large as . Therefore, we can account for the extra factor by inserting .

Common Mistakes

Choosing the wrong u

Do not automatically choose the most complicated-looking piece. Choose an expression whose derivative also appears in the integral. The best substitution is the one that makes the entire integral simpler.

Forgetting the differential

If you choose , immediately write down . The differential tells you exactly what must be replaced in the original integral.

Losing a constant factor

If but your integral contains , you must account for the factor of 3. Missing that constant produces an answer that looks reasonable but differentiates to the wrong function.

Forgetting to substitute back

For an indefinite integral, your final answer should normally be written in terms of , not . Once you integrate, replace the temporary variable with the original expression.

Confusing logarithmic integration with logarithmic differentiation

Remember that . But the integral of is not simply . The logarithm pattern requires the numerator to match the derivative of the denominator.

Practice Problems

Question 1

Evaluate .

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Question 2

Evaluate .

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Question 3

Evaluate .

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Question 4

Evaluate .

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Question 5 (Combined Skill)

Evaluate .

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