Problem 53 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. 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: Expand the Integrand

First, let's expand the expression inside the integral. The given function is \( \cos \theta (\tan \theta + \sec \theta) \). We'll distribute \( \cos \theta \) across the terms in parentheses: \( \cos \theta \cdot \tan \theta + \cos \theta \cdot \sec \theta \).

Step 2: Simplify Each Term

Now, simplify each term of the expanded integrand. We have: \( \cos \theta \cdot \tan \theta = \sin \theta \) (since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)) and \( \cos \theta \cdot \sec \theta = 1 \) (since \( \sec \theta = \frac{1}{\cos \theta} \)). Thus, the integrand simplifies to \( \sin \theta + 1 \).

Step 3: Integrate Simplified Terms

Integrate the simplified function term by term. Remember, the integral of \( \sin \theta \) is \(-\cos \theta + C_1\) and the integral of a constant \(1\) is \( \theta + C_2\). Therefore, the integral becomes: \(-\cos \theta + \theta + C\).

Step 4: Differentiate to Check the Work

Differentiate the result to ensure it matches the original integrand. The derivative of \(-\cos \theta + \theta + C\) is \( \sin \theta + 1\), which matches the simplified expression for the original integrand. Thus, the solution is correct.

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, is essentially the reverse process of differentiation. When we talk about the indefinite integral of a function, we're searching for a function whose derivative equates to the given function. In notation, if we have a function \( f(x) \), the indefinite integral is represented as \( \int f(x) \, dx \). Instead of a specific value, the result of an indefinite integral is a family of functions differing by a constant of integration, usually denoted by \( C \). This constant reflects that derivatives of constant terms are zero; hence when differentiating, the constant disappears.

• The integral \( \int \cos \theta \, d\theta \) yields \( \sin \theta + C \) since the derivative of \( \sin \theta \) results in \( \cos \theta \).
• Similarly, \( \int \theta \, d\theta \) results in \( \frac{\theta^2}{2} + C \).

Recognizing patterns and functions involved in an indefinite integral allows for retrieving the original function before differentiation.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that consist of trigonometric expressions. These integrals are a fundamental part of calculus, often requiring specific techniques to simplify and solve the expressions. Trigonometric functions such as sine, cosine, tangent, and their reciprocals are inserted into equations, and their integrals provide the antiderivatives. In our example, we are dealing with trigonometric functions \( \cos \theta \), \( \tan \theta \), and \( \sec \theta \). To solve these integrals efficiently:

• Simplify expressions whenever possible, as with the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Utilize identities to simplify the function into solvable parts, as seen when \( \cos \theta \cdot \sec \theta \) simplifies to \(1\).
• Once simplified, online formulas can be applied directly to common integrals, bypassing unnecessary steps.

Solving trigonometric integrals often requires combining simplifications with known identities to reach the desired solution.

Integration Techniques

Various integration techniques allow us to tackle diverse integrals effectively—each suited for different types of integrands. We've employed distribution and simplification techniques in our given problem. Here's how they work together:

• Distribution: By distributing \( \cos \theta \) across \( \tan \theta + \sec \theta \), the integral is converted into simpler terms that can be integrated separately, which is \( \cos \theta \cdot \tan \theta + \cos \theta \cdot \sec \theta \).
• Simplification: Identifying that \( \cos \theta \cdot \tan \theta = \sin \theta \) and \( \cos \theta \cdot \sec \theta = 1 \) further turns the problem into manageable parts, namely \( \sin \theta + 1 \).
• Term-wise Integration: After simplification, the individual parts are integrated separately using known results. For example, \( \int \sin \theta \, d\theta \) and \( \int 1 \, d\theta \) are computed individually, yielding the final antiderivative.

These techniques are powerful tools to manage different integrals, especially those incorporating trigonometric functions or requiring algebraic manipulation.