Problem 8 - Essential Calculus Early Transcendentals

Question

Find the most general antiderivative of the function (Check your answer by differentiation $$r(\theta)=\sec \theta \tan \theta-2 e^{\theta}$$

Step-by-Step Solution

Answer

The general antiderivative is \(\sec \theta - 2e^{\theta} + C\).

Step 1: Identify the Basic Antiderivatives

The given function is \( r(\theta) = \sec \theta \tan \theta - 2e^{\theta} \). We need to find the antiderivatives of \( \sec \theta \tan \theta \) and \( e^{\theta} \) separately. Recall that the antiderivative of \( \sec \theta \tan \theta \) is \( \sec \theta + C_1 \), and the antiderivative of \( e^{\theta} \) is \( e^{\theta} + C_2 \).

Step 2: Solve Antiderivative for Each Part

For the function \( \sec \theta \tan \theta \), the antiderivative is:\[ \int \sec \theta \tan \theta \, d\theta = \sec \theta + C_1 \].For the function \( -2e^{\theta} \), using the constant multiple rule, the antiderivative is:\[ \int -2 e^{\theta} \, d\theta = -2e^{\theta} + C_2 \].

Step 3: Combine Results for the General Antiderivative

Combine both antiderivatives computed in Step 2 to get the general antiderivative of \( r(\theta) \):\[ \int r(\theta) \, d\theta = \sec \theta - 2e^{\theta} + C \].Here, \( C = C_1 + C_2 \), represents the constant of integration.

Step 4: Differentiate to Verify

Differentiate the combined result to verify our antiderivative:\[ \frac{d}{d\theta} \left( \sec \theta - 2e^{\theta} + C \right) = (\sec \theta \tan \theta) - 2e^{\theta} = r(\theta) \].Since this differentiation gives us back the original function \( r(\theta) \), it confirms our solution is correct.

Key Concepts

Integration Techniques

Antiderivatives are a fundamental part of calculus, focusing on the reverse process of differentiation. The goal of integration is to find a function whose derivative equals a given function.
Integration techniques help in solving these problems by providing methods to efficiently find antiderivatives. In this exercise, we look at finding antiderivatives of specific functions.

• **Recognize Standard Forms**: Many functions have known antiderivatives, and recognizing these forms is key. For example, the antiderivative of \(\sec \theta \tan \theta\) is known directly as \(\sec \theta\), due to its derivative properties.
• **Constant Multiple Rule**: When a function is multiplied by a constant, this rule states that the antiderivative of the constant times a function is the constant times the antiderivative of the function.
  Thus, the antiderivative of \(-2e^{\theta}\) relies on the antiderivative of \(e^{\theta}\), which is \(e^{\theta}\) itself.
• **Combining Results**: After finding individual antiderivatives, they are combined to give the full solution, plus a constant \(C\), derived from adding individual constants of integration.

By mastering these techniques, you can confidently solve similar calculus problems with ease. Remember, if you need to, always go back to each part, and tackle it separately for clarity before combining.

Differentiation

Checking the correctness of an antiderivative solution is essential, and this is where differentiation comes into play.
Differentiation is the mathematical process of finding the derivative of a function, which measures the rate at which a function changes at any point.
This inverse relationship with integration means we can verify our answers by differentiating antiderivatives.
When given \( \sec \theta - 2e^{\theta} + C \) as a solution:

• **Differentiate Each Term**: + Differentiate \(\sec \theta\) to get \(\sec \theta \tan \theta\); differentiating \(-2e^{\theta}\) results in \(-2e^{\theta}\).
• **Constant Rule**: + Differentiating the constant \(C\) results in zero, as constants do not change when derived.
• **Combine Results**: + By adding the differentiated terms, \(\sec \theta \tan \theta - 2e^{\theta}\), we confirm that it matches the original function, verifying the correctness of our antiderivative.

By systematically checking, differentiation not only reassures us but also solidifies our understanding of the integral's correctness. This ensures a deep comprehension of how particular functions behave and relate differently.

Transcendental Functions

Transcendental functions are non-algebraic functions that include elements such as exponential and trigonometric functions.
These functions do not have solutions in terms of simple polynomial equations.
Understanding their behavior is crucial in calculus and higher-level mathematics.
In our exercise, both \(\sec\theta\) and \(e^{\theta}\) are transcendental functions.

• **Exponential Function**: + The function \(e^{\theta}\) is classic for exhibiting continuous growth. Differentiating or integrating \(e^{\theta}\) does not change its form, showcasing its unique property and making calculations straightforward.
• **Trigonometric Functions**: + The \(\sec\theta\) function, with its derivative \(\sec\theta \tan\theta\), provides a prime example of the trigonometric realm. It is necessary to recognize these derivatives quickly when integrating or differentiating trigonometric functions.

Handling transcendental functions requires familiarity with their unique properties—how they differentiate and integrate, and how they fundamentally vary from algebraic functions. Comprehensive practice ensures these become second nature in solving calculus problems.