u-Substitution with Trigonometric Functions

What u-Substitution Is For

u-substitution is the integration technique that reverses the chain rule. It is especially useful when a trigonometric function contains another function inside it, such as , or when a trigonometric expression is multiplied by the derivative of that inside function.

The basic idea is simple: replace the complicated inside expression with a new variable, . This temporarily turns a difficult integral into a familiar one.

For example, consider:

The expression is the inside of the cosine, and its derivative is . That is exactly the pattern we want for u-substitution.

The Formula

Suppose . Then:

For integration, the corresponding substitution idea is:

In practice, you choose to be the expression inside the trigonometric function, differentiate it to obtain , and rewrite the entire integral in terms of .

How to Recognize a Trigonometric u-Substitution

The most important skill is recognizing the pattern. Look for a trigonometric function containing an expression that has its derivative sitting nearby.

Sine: If you see , look for .
Cosine: If you see , look for .
Tangent: If you see , look for .
Secant: If you see , look for .

The derivative does not have to appear exactly. Sometimes it is present up to a constant multiple. If that happens, you can factor the constant out or compensate for it after the substitution.

A Step-by-Step Method

Choose u. Usually, choose the expression inside the trigonometric function.
Differentiate u. Find .
Rewrite the differential. Solve for and substitute it into the integral.
Rewrite the trigonometric function. Replace its inside expression with .
Integrate with respect to u. The goal is to obtain a basic integral.
Substitute back. Replace with the original expression.

Worked Examples

Example 1: Sine with a Linear Inside Function

Evaluate:

Step 1: Choose the inside expression:

Step 2: Differentiate:

Step 3: Solve for :

Step 4: Substitute:

Step 5: Integrate:

Result:

Example 2: Cosine with a Polynomial Inside Function

Evaluate:

Step 1: Choose:

Step 2: Differentiate:

Step 3: Rewrite the differential:

Step 4: Substitute:

Step 5: Integrate:

Result:

Example 3: A Constant Multiple of the Derivative

Evaluate:

Here, the derivative of is . The integral contains , which is four times the derivative we need.

Step 1: Choose:

Step 2: Then:

Step 3: Solve for :

Step 4: Substitute:

Step 5: Integrate and substitute back:

Result:

Example 4: Tangent with Its Derivative

Evaluate:

Step 1: Choose:

Step 2: Differentiate:

Step 3: Notice that the integral contains exactly . Therefore:

Step 4: Integrate:

Result:

Example 5: Secant Squared

Evaluate:

Step 1: Choose:

Step 2: Differentiate:

Step 3: Substitute:

Step 4: Integrate:

Result:

Why This Works: u-Substitution Is the Chain Rule Backward

There is a useful way to understand u-substitution rather than simply memorizing it.

The chain rule tells us that:

Integration reverses differentiation. Therefore, if an integrand contains the structure , we can recognize it as the derivative of .

That is why an integral such as:

becomes easy after setting . The complicated composition disappears, leaving the basic integral of .

Common Mistakes

Choosing the wrong expression for u

The best choice for is usually the expression inside the trigonometric function. If you choose something that does not simplify the integral, the substitution will not help.

Forgetting the constant multiple

Suppose . If your integral contains , you need to account for the factor of 3. Otherwise your final answer will be off by a constant factor.

Forgetting to substitute back

For an indefinite integral, your final answer should normally be written in terms of , not . Once you finish integrating with respect to , replace it with the original expression.

Missing the derivative pattern

The derivative of the inside function does not always appear exactly. It may differ by a constant multiple. Always compare the expression you have with before deciding that u-substitution does not work.

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 (Constant Multiple)

Evaluate .

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The AP Calculus Takeaway

When you see a trigonometric integral on the AP Calculus exam, do not immediately assume that you need a complicated integration technique. First look for a function-and-derivative pair.

If you see a trigonometric function containing an expression such as , ask whether its derivative is also present somewhere in the integrand.

The mental checklist is:

What is inside the trig function?
What is its derivative?
Is that derivative present, perhaps up to a constant?
If yes, let the inside expression be u.
Integrate in terms of u.
Substitute back.

Once you recognize this pattern, many trigonometric integrals stop looking like integration problems and start looking like the chain rule in reverse.

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