Problem 38 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int 3 \cos 5 \theta d \theta $$

Step-by-Step Solution

Answer

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

Step 1: Recognize Integration Formula

The integral we are trying to solve is \( \int 3 \cos 5 \theta \, d\theta \). The standard formula for integrating \( \cos(kx) \) is \( \frac{1}{k} \sin(kx) + C \), where \( C \) is the constant of integration.

Step 2: Identify Constants and Functions

In the integral \( 3 \cos 5 \theta \, d\theta \), the constant multiplier is \( 3 \), and \( k = 5 \) from \( 5\theta \). The antiderivative will involve \( \frac{1}{5} \sin 5\theta \) due to the \( 5 \theta \) term.

Step 3: Apply the Integration Formula

Apply the integration formula: \( \int \cos 5 \theta \, d\theta = \frac{1}{5} \sin 5 \theta + C \). Multiplying by the constant \( 3 \), we have: \( 3 \times \frac{1}{5} \sin 5 \theta + C = \frac{3}{5} \sin 5 \theta + C \).

Step 4: Verification by Differentiation

Differentiate \( \frac{3}{5} \sin 5 \theta + C \) to verify. The derivative is \( \frac{3}{5} \times 5 \cos 5 \theta = 3 \cos 5 \theta \) which matches the original integrand. Thus, the solution is correct.

Key Concepts

Indefinite Integrals

Indefinite integrals are a key concept in calculus, used to find the antiderivative of a function. Unlike definite integrals, which give a numerical result, indefinite integrals yield a function or family of functions.

• The indefinite integral is represented by the symbol \( \int \), followed by the function and the differential.
• The result of an indefinite integral includes a constant of integration, \( C \), to account for all possible constant shifts in the antiderivative.
• In our exercise, we worked with the function \( 3 \cos 5\theta \) and sought its antiderivative.

To solve this, we apply integration techniques such as substitution or recognizing standard forms from integration tables. The key is to match the integrand to a known formula and manipulate accordingly, incorporating any constants as we proceed.

Trigonometric Integration

Trigonometric integration is a specialized technique for integrating functions involving trigonometric terms like sine and cosine. It requires familiarity with trigonometric identities and standard integration formulas.

• For \( \cos(kx) \), the integration formula is \( \frac{1}{k} \sin(kx) + C \).
• In our exercise, we had \( \cos 5\theta \) where \( k = 5 \), resulting in \( \frac{1}{5} \sin 5\theta \).
• The function \( 3 \cos 5\theta \) includes a constant multiplier, so we simply multiply the constant by the integral of \( \cos 5\theta \).

By following these steps, you can tackle many trigonometric integrals by identifying constants and applying known formulas fluidly.

Differentiation Verification

Once you have found an antiderivative, it is essential to verify its correctness by differentiation. If differentiating leads you back to the original function, your solution is correct.

• Differentiation reverses the integration process. It's your primary tool for verification of indefinite integrals.
• In differentiating the solution \( \frac{3}{5} \sin 5\theta + C \), we multiply by the inner derivative, function values, and simplify.
• The end result should give you \( 3 \cos 5\theta \), thus confirming the antiderivative is accurately computed.

This step is not just a formality. It ensures mathematical precision and provides confidence in your integration skills.