Problem 11 - Calculus - AP Edition

Question

Evaluate the integral. $$\int \sin ^{2} x \cos ^{2} x d x$$

Step-by-Step Solution

Answer

The integral is \( \frac{x}{8} - \frac{\sin(4x)}{32} + C \).

Step 1: Use a Trigonometric Identity

Notice that \( \sin^2(x) \cos^2(x) \) can be rewritten using the trigonometric identity for the product: \( \sin^2(x) \cos^2(x) = \frac{1}{4} \sin^2(2x) \). This allows you to simplify the integral to \( \int \frac{1}{4} \sin^2(2x) \, dx \).

Step 2: Use Another Trigonometric Identity

The function \( \sin^2(2x) \) can be further simplified using the identity: \( \sin^2(2x) = \frac{1 - \cos(4x)}{2} \). Substitute this into the integral to get \( \int \frac{1}{4} \cdot \frac{1 - \cos(4x)}{2} \, dx \), which simplifies to \( \frac{1}{8} \int (1 - \cos(4x)) \, dx \).

Step 3: Integrate the Simplified Expression

Now, integrate the expression \( \frac{1}{8} \int (1 - \cos(4x)) \, dx \). This results in \( \frac{1}{8} \int 1 \, dx - \frac{1}{8} \int \cos(4x) \, dx \). By integrating separately, you get \( \frac{1}{8} x - \frac{1}{32} \sin(4x) + C \) for the second part using basic integration rules.

Step 4: Combine and Simplify

Combine the expressions from the integration to arrive at the final answer. Therefore, the integral evaluates to \( \frac{x}{8} - \frac{\sin(4x)}{32} + C \).

Key Concepts

Trigonometric Identities

Trigonometric identities are useful tools that allow us to manipulate and simplify expressions involving trigonometric functions. In our specific problem, the trigonometric identity

• \( \sin^2(x) \cos^2(x) = \frac{1}{4} \sin^2(2x) \)

was used to transform a product of sine and cosine squares into a single trigonometric function. This reduces complexity and makes integration more straightforward. Additionally, we applied another identity:

• \( \sin^2(2x) = \frac{1 - \cos(4x)}{2} \)

This identity rewrites the square of a sine function in terms of cosine, which is easier to integrate. Recognizing and using these identities effectively can transform a challenging integral into a manageable one. When you encounter products or higher powers of trigonometric functions, remember to consider these identities to simplify the expression.

Definite Integrals

Definite integrals are used to find the area under a curve over a specific interval. However, in this exercise, we focus primarily on indefinite integrals because we are not evaluating between bounds. The principles still relate here, as substituting trigonometric identities simplifies the problem to a more straightforward expression, which could be computed as a definite integral if bounds were provided. In practice:

• The process involves identifying expressions that match an identity.
• Substitute and simplify the expression.
• Perform the integration using basic rules.

For a definite integral, additional steps to evaluate limits are required, which involves assessing the antiderivative at the upper and lower limits and finding their difference. While our example is indefinite, understanding definite integrals prepares you for such computations.

Basic Integration Rules

Basic integration rules are foundational for solving integrals and provide the guidelines for integrating simple functions. In the exercise, once we simplified the integrand using trigonometric identities, we applied basic integration rules:

• For \( \int 1 \, dx \), the rule is simply \( x + C \).
• For \( \int \cos(ax) \, dx \), the integral becomes \( \frac{1}{a} \sin(ax) + C \).

These rules allow for direct integration once the integrand is simplified. The constant \( C \) represents an arbitrary constant of integration, crucial for indefinite integrals to account for all possible antiderivatives. By using these straightforward rules, integration becomes a manageable process, turning more complex problems into simple calculations once initial simplifications are made.