Problem 55 - 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. $$\int \cos \theta(\tan \theta+\sec \theta) d \theta$$

Step-by-Step Solution

Answer

The antiderivative is \(-\cos \theta + \theta + C\).

Step 1: Distribute the Cosine Function

First, distribute \( \cos \theta \) across \( \tan \theta + \sec \theta \). This gives us:\[\cos \theta \cdot \tan \theta + \cos \theta \cdot \sec \theta.\]We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Substituting these, we get:\[\cos \theta \cdot \tan \theta = \sin \theta \quad \text{and} \quad \cos \theta \cdot \sec \theta = 1.\]Thus, the integrand simplifies to \( \sin \theta + 1 \).

Step 2: Integrate using Basic Rules

Now that the integrand is \( \sin \theta + 1 \), integrate each term individually:\[\int (\sin \theta + 1) \, d\theta = \int \sin \theta \, d\theta + \int 1 \, d\theta.\]The integral of \( \sin \theta \) is \(-\cos \theta \) and the integral of \( 1 \) is \( \theta \). Therefore, the antiderivative is:\[-\cos \theta + \theta + C,\]where \( C \) is the constant of integration.

Step 3: Verify by Differentiation

To ensure our solution is correct, differentiate \(-\cos \theta + \theta + C\) with respect to \( \theta \):\[\frac{d}{d\theta}(-\cos \theta + \theta + C) = \sin \theta + 1.\]Since this matches the simplified original integrand, our antiderivative \(-\cos \theta + \theta + C\) is verified as correct.

Key Concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative equals the original function. When we integrate, we're essentially finding a function that, when differentiated, gives us the integrand. In the given exercise, we started with the integrand \( \cos \theta(\tan \theta + \sec \theta) \). After simplifying, it becomes \( \sin \theta + 1 \), which is easier to integrate. The integral of \( \sin \theta \) is \(-\cos \theta \), and the integral of \( 1 \) is \( \theta \). Thus, the antiderivative is \(-\cos \theta + \theta + C\), where \( C \) is a constant of integration. This constant represents the family of functions differing only by a constant. We verify this result by differentiation, ensuring the derivative of our found antiderivative returns us to the original integrand. Finding antiderivatives is essential in calculus, as it forms the basis for solving more complex integration problems. It helps us understand how rates of change accumulate over time or space.

Trigonometric Integration

Trigonometric integration involves integrating functions that include trigonometric identities and expressions. These functions often appear in real-world applications like physics and engineering. In this exercise, the given function is a product of trigonometric functions \( \cos \theta \), \( \tan \theta \), and \( \sec \theta \). During the integration process, we used trigonometric identities:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)

These identities help further simplify the integrand, turning it into a more manageable form of \( \sin \theta + 1 \). Trigonometric integration often requires using these identities to reduce complex expressions into simpler integrable forms. Understanding these identities and how to manipulate them is crucial for successful integration.

Integration by Substitution

Integration by substitution is a method used to simplify complex integrals, especially when direct integration is challenging. Although not directly applied in our given solution, it's a complementary technique that can be useful when trigonometric identities and simpler forms are not enough. The general idea is to substitute a part of the integrand with a new variable, making the integral easier to solve. This is akin to using the chain rule in reverse for differentiation. For instance, if dealing with an integral that includes a composite function, substitution can transform the integral into a more straightforward form. You introduce a new variable \( u \), set it equal to a part of the integrand, and replace the remaining variables in terms of \( u \). A typical step-by-step would be:

• Select a portion of the integrand to substitute and let it equal \( u \).
• Differentiate \( u \) to find \( du \).
• Replace the original variables and differential with \( u \) and \( du \).
• Integrate with respect to \( u \).
• Substitute back the original variables to find the antiderivative.

This method simplifies integration, allowing us to integrate more complex functions effectively. As you grow more familiar with trigonometric identities and substitutions, you'll be better prepared to tackle a broader range of integrals.