Problem 45 - University Calculus: Early Transcendentals

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. $$\int 7 \sin \frac{\theta}{3} d \theta$$

Step-by-Step Solution

Answer

The most general antiderivative is \(-21 \cos \left(\frac{\theta}{3}\right) + C\).

Step 1: Identify the Integral

We begin with finding the antiderivative of the function \(7 \sin\left(\frac{\theta}{3}\right)\), where the integral is \(\int 7 \sin\left(\frac{\theta}{3}\right) d \theta\).

Step 2: Make a Substitution

Since the argument of the sine function is \(\frac{\theta}{3}\), we use a substitution method. Let \(u = \frac{\theta}{3}\). Thus, \(\frac{du}{d\theta} = \frac{1}{3}\) or \(d\theta = 3du\).

Step 3: Substitute and Simplify the Integral

Substitute \(u\) and \(d\theta = 3du\) into the integral: \(\int 7 \sin u \cdot 3 du = 21 \int \sin u \, du\).

Step 4: Integrate with Respect to u

The integral of \(\sin u\) is \(-\cos u\). Therefore, \(21 \int \sin u \, du = 21 [-\cos u] + C = -21 \cos u + C\), where \(C\) is the constant of integration.

Step 5: Re-substitute to Original Variable

Replace \(u\) with \(\frac{\theta}{3}\) to get the antiderivative in terms of \(\theta\): \(-21 \cos \left(\frac{\theta}{3}\right) + C\).

Step 6: Differentiate to Confirm Solution

Differentiate \(-21 \cos \left(\frac{\theta}{3}\right) + C\) with respect to \(\theta\). This gives \(21 \cdot \sin\left(\frac{\theta}{3}\right) \cdot \frac{1}{3} = 7 \sin\left(\frac{\theta}{3}\right)\), which matches the original integrand, confirming the solution is correct.

Key Concepts

Substitution Method

The substitution method is a powerful technique in solving integrals, especially when dealing with functions that contain a composition of expressions, like trigonometric ones. The idea is to simplify the integral by substituting a part of the integrand with a new variable, usually denoted as \( u \).

This new variable should ideally simplify the integral into a basic form that is easy to solve. Here's how you can apply it:

• Identify a portion of the expression that, when substituted, would simplify the integral.
• Rewrite the expression in terms of the new variable \( u \).
• Replace \( d\theta \) or the differential term with the corresponding \( du \).
• Integrate the new function in terms of \( u \).
  
• Re-substitute the original variable back into the solution.
  

The goal is to make the integral more approachable, transforming a complex function into a simple one.

Antiderivative

An antiderivative, also known as an indefinite integral, is the reverse process of differentiation. It represents the collection of all functions whose derivative gives the original function back.

When computing an indefinite integral, you must include a constant \( C \) at the end. This accounts for the family of functions that differ by a constant but still have the same derivative.

• Given a function \( f(x) \), its antiderivative is represented as \( \int f(x) \, dx \).
• The general solution will include the constant \( C \), which can take any real number value.
• This concept is crucial in determining the most general form of functions after integration.
  

By recognizing common forms of derivatives, you can identify their antiderivatives more easily.

Differentiation

Differentiation is the process of finding the derivative of a function. It measures how a function changes as its input changes. In the context of verifying an antiderivative, differentiation plays a key role.

After computing the indefinite integral, differentiating it should yield the original integrand. Here's why it's important:

• Confirming correct integration: By differentiating the antiderivative, you can check if the process was done correctly by comparing the result to the initial integrand.
• Gaining insight to check errors: Differentiation helps identify mistakes in integration steps.
• Understanding function behavior: Derivatives indicate rates of change, providing insights into the function's behavior.
  

Thus, differentiation verifies that your solution is both accurate and consistent with the original function.

Trigonometric Integrals

Trigonometric integrals involve integrands that are trigonometric functions such as sine, cosine, and tangent. These integrals often require specific techniques or knowledge of fundamental trigonometric identities to solve effectively.

For instance, the integral of \( \sin u \) results in \(-\cos u\), and vice versa. Key steps in solving trigonometric integrals include:

• Recognizing basic trigonometric forms or identities that simplify the problem.
• Using substitution to handle more complex or nested trigonometric expressions.
• Applying symmetry or periodicity of trigonometric functions to simplify evaluations.
  

By mastering these concepts, you can handle trigonometric integrals with confidence and efficiency.