Problem 69 - 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. $$\int \cos \theta(\tan \theta+\sec \theta) d \theta$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Integrand

First, let's simplify the expression inside the integral. We have \( \int \cos \theta (\tan \theta + \sec \theta) d\theta \). Begin by distributing \( \cos \theta \), resulting in two terms: \( \cos \theta \tan \theta \) and \( \cos \theta \sec \theta \). Thus, the integral becomes: \( \int (\cos \theta \tan \theta + \cos \theta \sec \theta) d\theta \).

Step 2: Simplify Each Term

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Using these identities: \( \cos \theta \tan \theta = \sin \theta \) and \( \cos \theta \sec \theta = 1 \). So, the integral becomes \( \int (\sin \theta + 1) d\theta \).

Step 3: Find the Antiderivative

Now, integrate each term separately: \( \int \sin \theta \, d\theta + \int 1 \, d\theta \). The antiderivative of \( \sin \theta \) is \(-\cos \theta \), and the antiderivative of \( 1 \) is \( \theta \). Thus, the antiderivative of the entire expression is \(-\cos \theta + \theta + C \), where \( C \) is the constant of integration.

Step 4: Check by Differentiation

Differentiate \(-\cos \theta + \theta + C \) to verify. The derivative of \(-\cos \theta \) is \( \sin \theta \), and the derivative of \( \theta \) is \( 1 \), while the derivative of a constant \( C \) is \( 0 \). Therefore, the derivative of the antiderivative is \( \sin \theta + 1 \), which matches the simplified form of the original integrand. This confirms our solution is correct.

Key Concepts

Antiderivative

Finding an antiderivative essentially means reversing the process of differentiation. The challenge is to determine a function whose derivative will yield the given function. This unique function is known as the most general antiderivative because it includes a constant of integration, denoted as \( C \).

This constant is necessary because when you differentiate a constant, it disappears, meaning there are infinitely many functions that differ only by a constant which have the same derivative. In essence, every function has an antiderivative, which adds flexibility in definite integration and is the core of finding indefinite integrals.

In the context of our solution, the antiderivative of the function \( \cos \theta(\tan \theta + \sec \theta) \) was determined by simplifying it to \( \sin \theta + 1 \), then integrating each component. This results in \(-\cos \theta + \theta + C \). Verifying by differentiation ensures the antiderivative is correct, as differentiation should yield the original function.

Trigonometric Identities

Trigonometric identities are crucial tools in calculus and trigonometry that simplify expressions and solve equations more easily. These identities allow us to rewrite trigonometric functions in different but equivalent forms.

Important basic identities include:

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

In our exercise, we used these identities to transform the original integrand \( \cos \theta(\tan \theta + \sec \theta) \). By recognizing these fundamental relationships, we simplified each term: \( \cos \theta \tan \theta = \sin \theta \) and \( \cos \theta \sec \theta = 1 \).

Applying trigonometric identities in integration tasks is beneficial, as they can help simplify the integrand, making the integration process more straightforward. This process is often a critical step before finding antiderivatives.

Integration Techniques

Integration techniques are methods developed to handle a wide variety of integrals. Each technique has its own set of rules and applicable scenarios. Fundamental techniques include:

• Simplification of the integrand using algebraic manipulation
• Application of basic antiderivative rules
• Usage of trigonometric identities

The exercise utilized the simplification technique by distributing \( \cos \theta \) over \( \tan \theta + \sec \theta \) and then simplifying further using trigonometric identities. This reduced the complexity of integration from handling a product of functions to simple polynomial integration.

Ultimately, integration techniques empower us to solve more complex problems by transforming them into solvable parts. Every integral can be approached methodically by exploring different techniques until the simplest form is achieved for the antiderivative to be calculated easily.