Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \cos ^{3} 3 \theta \sin ^{-2} 3 \theta d \theta $$

Step-by-Step Solution

Answer

The integral is \( -\frac{1}{3} \cot 3\theta - \frac{1}{3} \sin 3\theta + C \).

Step 1: Simplify the Expression

First, recognize that \( \cos^3 3\theta \sin^{-2} 3\theta \) can be simplified by writing it as \( \frac{\cos^3 3\theta}{\sin^2 3\theta} \). To simplify further, express it as \( \cos 3\theta \left(\frac{\cos^2 3\theta}{\sin^2 3\theta}\right) = \cos 3\theta \cot^2 3\theta \).

Step 2: Use Trigonometric Identity

Use the trigonometric identity \( \cot^2 x = \csc^2 x - 1 \) to rewrite \( \cot^2 3\theta \) as \( \csc^2 3\theta - 1 \). This gives us the expression \( \cos 3\theta (\csc^2 3\theta - 1) \).

Step 3: Distribute and Integrate

Distribute the \( \cos 3\theta \) through the expression, yielding \( \cos 3\theta \csc^2 3\theta - \cos 3\theta \). Now, integrate each term separately: 1. \( \int \cos 3\theta \csc^2 3\theta \, d\theta \) 2. \( \int -\cos 3\theta \, d\theta \)

Step 4: Integrate First Term Using Substitution

For \( \int \cos 3\theta \csc^2 3\theta \, d\theta \), use substitution. Let \( u = \sin 3\theta \). Consequently, \( du = 3\cos 3\theta \, d\theta \). Then, \[ \int \cos 3\theta \csc^2 3\theta \, d\theta = \frac{1}{3} \int \csc^2 3\theta \, du = \frac{1}{3} (-\cot u) + C_1 = -\frac{1}{3} \cot 3\theta + C_1 \].

Step 5: Integrate Second Term

For \( \int -\cos 3\theta \, d\theta \), the integral is straightforward: \[ \int -\cos 3\theta \, d\theta = -\frac{1}{3} \sin 3\theta + C_2 \].

Step 6: Combine Results

Combine the integrated results and simplify, \[ -\frac{1}{3} \cot 3\theta - \frac{1}{3} \sin 3\theta + C \].Combine the constants \( C_1 \) and \( C_2 \) into a single constant \( C \).

Step 7: Final Answer

The integral of the given function is \[ -\frac{1}{3} \cot 3\theta - \frac{1}{3} \sin 3\theta + C \].

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express one trigonometric function in terms of others. They allow us to simplify complex trigonometric expressions, making integration and other calculations easier. One crucial identity used in the integral solution is the Pythagorean identity involving the cotangent and cosecant functions:

• \( \cot^2 x = \csc^2 x - 1 \)

This identity helps transform \( \cot^2 3\theta \) into \( \csc^2 3\theta - 1 \), simplifying the integration process. By using these identities, we reduce the complexity of the trigonometric expression, revealing relationships that are not immediately obvious but are essential for solving integrals effectively.

Substitution Method in Integration

The substitution method in integration is a powerful technique that simplifies the integration process by transforming a complex integral into a more manageable form. In the exercise, we perform the substitution by letting \( u = \sin 3\theta \), which implies \( du = 3\cos 3\theta \, d\theta \).
This approach turns the integral \( \int \cos 3\theta \csc^2 3\theta \, d\theta \) into a simpler form that can be directly integrated using known forms.Here's how substitution helps:

• Converts the problem into a standard integral form.
• Simplifies complex trigonometric expressions.
• Facilitates the use of basic integral tables.

By applying substitution, the integration becomes straightforward, and we can solve it using elementary functions.

Cotangent Function

The cotangent function, denoted as \( \cot x \), is an essential trigonometric function frequently encountered in calculus. It is the reciprocal of the tangent function and is defined as \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).
In trigonometric integration, the cotangent function often appears in situations where simplification involves division of sine and cosine functions.Key properties of the cotangent function:

• Periodicity — it repeats every \( \pi \) radians.
• Symmetry — it is an odd function, \( \cot(-x) = -\cot x \).
• Useful in identities like \( \cot^2 x = \csc^2 x - 1 \).

The cotangent often serves as a bridge, allowing us to translate between rational expressions involving sine and cosine to more solvable forms, as seen in this integration exercise.

Cosecant Function

The cosecant function, symbolized as \( \csc x \), is the reciprocal of the sine function and is defined by \( \csc x = \frac{1}{\sin x} \). It is less commonly used on its own but plays a critical role in trigonometric identities and integrals.
When integrating with trigonometric expressions, the cosecant function provides opportunities to employ identities and techniques that are crucial for simplification. Important aspects of the cosecant function:

• It is undefined where \( \sin x = 0 \) due to division by zero.
• Periodicity makes it repeat every \( 2\pi \) radians.
• In integrals, it often complements identities for cotangent and tangent.

Using \( \csc 3\theta \) in trigonometric identities like \( \cot^2 3\theta = \csc^2 3\theta - 1 \) allows us to rewrite expressions in forms amenable to straightforward integration, facilitating the solution of complex integrals.