Problem 38 - Thomas Calculus in SI Units

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int 3 \cos 5 \theta d \theta$$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{3}{5} \sin 5 \theta + C \).

Step 1: Identify the Integral's Basic Form

The integral provided is \( \int 3 \cos 5 \theta \, d\theta \). Notice that it has the basic form \( \int a \cos kx \, dx \), where \( a = 3 \) and \( k = 5 \).

Step 2: Write the Antiderivative Formula

The antiderivative of \( \cos kx \) is \( \frac{1}{k} \sin kx + C \), where \( C \) is the constant of integration. Here the constant multiplier \( k \) appears in the denominator.

Step 3: Compute the Indefinite Integral

Apply the formula for the antiderivative of \( \cos 5 \theta \): \[ \int 3 \cos 5 \theta \, d\theta = 3 \cdot \left( \frac{1}{5} \sin 5 \theta \right) + C = \frac{3}{5} \sin 5 \theta + C \]

Step 4: Differentiate to Check the Solution

Differentiate the function \( \frac{3}{5} \sin 5 \theta + C \) to verify correctness: \[ \frac{d}{d\theta} \left( \frac{3}{5} \sin 5 \theta + C \right) = \frac{3}{5} \cdot 5 \cos 5 \theta = 3 \cos 5 \theta \] This matches the original integrand, so the solution is correct.

Key Concepts

Understanding Indefinite Integrals

An indefinite integral, often referred to as an antiderivative, is a fundamental concept in calculus. It is represented by the symbol \( \int \) and is used to find a function that, when differentiated, returns the integrand. Unlike definite integrals, indefinite integrals do not have boundaries or limits.
Instead, they include an arbitrary constant, \( C \), because differentiation of a constant is zero and it disappears during the process.
So, the family of functions represented by an indefinite integral includes all possible antiderivatives, varying by this constant.

• Expression: \( \int f(x) \, dx = F(x) + C \)
• Meaning: Finds a function \( F(x) \) such that \( F'(x) = f(x) \).
• Includes: An arbitrary constant \( C \).

The primary purpose of finding indefinite integrals is to solve differential equations or evaluate changes involving continuous functions.

Integration Technique for Trigonometric Functions

In dealing with integrals, especially those involving trigonometric functions like \( \cos \), it's beneficial to understand specific integration techniques. For the example \( \int 3 \cos 5\theta \, d\theta \), we recognize a common form involving a trigonometric identity.
Applying integration techniques involves identifying a suitable antiderivative based on the structure of the function. In our case, the rule for integrating \( \cos(kx) \) is using the antiderivative \( \frac{1}{k} \sin(kx) + C \).
While integrating, constants like \( 3 \) can be factored out of the integral, simplifying the expression.

• Identify the form of the integrand: \( a \cos(kx) \).
• Factor out constants: \( \int a \cos(kx) \, dx = a \int \cos(kx) \, dx \).
• Use known antiderivatives: \( \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \).

These systematic steps guide solving integrals, making them more approachable and manageable irrespective of the function's complexity.

Differentiation as a Check for Integration

Differentiation is the reverse process of integration and is typically used to verify the correctness of an indefinite integral solution. Once you have computed the antiderivative, differentiating it should yield the original integrand.
For our solved integral \( \frac{3}{5} \sin 5 \theta + C \), differentiating this function with respect to \( \theta \) involves applying the chain rule.
Differentiation helps confirm that no steps were missed in finding the correct antiderivative and the integral indeed works backward to the original function.

• Apply differentiation: \( \frac{d}{d\theta} F(\theta) = f(\theta) \).
• Ensure it matches: Return to integrand \( f(\theta) \).
• Verify continuity: Ensuring the transition between integration and differentiation is smooth.

This check-step is crucial in practice, as it builds confidence and assures exactitude in calculus-related problems, safeguarding us from errors.