AP Calculus AB 6.14 Integrating by Simplifying Trig Functions First

### AP Calculus AB 6.14: Integrating by Simplifying Trig Functions First

In this section, we focus on integrating trigonometric functions by first simplifying them, making the integration process easier. Trigonometric identities are often used to transform functions into a more manageable form before integrating.

#### Problem Overview
When faced with an integral involving trigonometric functions, you can apply trigonometric identities to simplify the integrand. This can lead to integrals that are easier to compute.

#### Key Trigonometric Identities
Here are a few key identities that can be useful for simplification:

1. **Pythagorean Identities:**
- \( \sin^2(x) + \cos^2(x) = 1 \)
- \( 1 + \tan^2(x) = \sec^2(x) \)
- \( 1 + \cot^2(x) = \csc^2(x) \)

2. **Double Angle Identities:**
- \( \sin(2x) = 2\sin(x)\cos(x) \)
- \( \cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x) \)

#### Steps for Integration

1. **Identify the Integral:**
- Start with an integral that involves trigonometric functions. For example:
\[
\int \sin^2(x) \cos(x) \, dx
\]

2. **Simplify Using Identities:**
- Use a trigonometric identity to simplify the integrand if possible. In this case, you can recognize that:
\[
\sin^2(x) = 1 - \cos^2(x)
\]
This identity can be used, or alternatively, you can directly substitute \( u = \cos(x) \) for integration.

3. **Substitution:**
- Choose an appropriate substitution. Here, let:
\[
u = \cos(x) \implies du = -\sin(x) \, dx
\]
Rearranging gives:
\[
\sin(x) \, dx = -du
\]
Therefore, we can rewrite the integral:
\[
\int \sin^2(x) \cos(x) \, dx = -\int (1 - u^2) u \, du
\]

4. **Integrate:**
- Expand and integrate:
\[
-\int (u - u^3) \, du = -\left( \frac{u^2}{2} - \frac{u^4}{4} \right) + C
\]
Substitute back for \( u \):
\[
-\left( \frac{\cos^2(x)}{2} - \frac{\cos^4(x)}{4} \right) + C
\]

5. **Simplify Final Result:**
- Distributing the negative sign:
\[
\frac{\cos^4(x)}{4} - \frac{\cos^2(x)}{2} + C
\]

#### Example

Consider the integral:
\[
\int \sin(x) \cos(x) \, dx
\]

**Step 1: Identify the Integral**
- Recognize the integral can be simplified.

**Step 2: Use a Trigonometric Identity**
- Recall that:
\[
\sin(2x) = 2\sin(x)\cos(x) \implies \sin(x)\cos(x) = \frac{1}{2}\sin(2x)
\]

**Step 3: Rewrite the Integral**
- The integral becomes:
\[
\int \sin(x) \cos(x) \, dx = \frac{1}{2} \int \sin(2x) \, dx
\]

**Step 4: Integrate**
- The integral of \( \sin(2x) \) is:
\[
-\frac{1}{2} \cos(2x) + C
\]

**Step 5: Combine Results**
- Thus:
\[
\int \sin(x) \cos(x) \, dx = -\frac{1}{4} \cos(2x) + C
\]

#### Conclusion
Simplifying trigonometric functions before integration allows you to leverage trigonometric identities, making the integration process smoother and more intuitive. This technique is essential in AP Calculus AB, providing a foundation for solving more complex integrals involving trigonometric functions.

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Norristown Area High School
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