Problem 50 - 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. $$\int \frac{2}{5} \sec \theta \tan \theta d \theta$$

Step-by-Step Solution

Answer

The most general antiderivative is \( \frac{2}{5} \sec \theta + C \).

Step 1: Identify the Integral Form

The integral presented is \( \int \frac{2}{5} \sec \theta \tan \theta \, d\theta \). This fits the pattern \( \int \sec x \tan x \, dx \), which has a standard result for its antiderivative.

Step 2: Use Known Antiderivative Rule

The antiderivative of \( \sec x \tan x \) is \( \sec x + C \), where \( C \) is a constant of integration. Thus, the antiderivative of \( \frac{2}{5} \sec \theta \tan \theta \) is \( \frac{2}{5} \sec \theta + C \).

Step 3: Differentiate to Check Answer

To verify the result, differentiate \( \frac{2}{5} \sec \theta + C \). The derivative of \( \frac{2}{5} \sec \theta \) is \( \frac{2}{5} \sec \theta \tan \theta \), and the derivative of a constant \( C \) is zero, confirming our solution is correct.

Key Concepts

Antiderivatives

An antiderivative is essentially the reverse of taking a derivative. When we have a function and we're looking for the antiderivative, we're trying to find the original function that would give us our given function after differentiation. In the context of indefinite integrals, we're adding up these infinite little slices under the curve to find a bigger function.In our original exercise, the function was \( \frac{2}{5} \sec \theta \tan \theta \). The antiderivative of \( \sec \theta \tan \theta \) is well-known in calculus, which provides a handy rule: the antiderivative is \( \sec \theta + C \), where \( C \) is the constant of integration. For \( \frac{2}{5} \sec \theta \tan \theta \), you simply scale this result by \( \frac{2}{5} \), giving you \( \frac{2}{5} \sec \theta + C \).To find the most general antiderivative, don't forget the "+ C." This represents any constant that could have been lost in the differentiation process.

Trigonometric Integrals

Trigonometric integrals involve integrals of combinations of sine, cosine, secant, and tangent functions. These integrals require familiarity with different identities and standard results, as trigonometric functions often simplify integration.For example, in this problem, knowing that the antiderivative of \( \sec x \tan x \) is \( \sec x \) is critical. This allows you to integrate without additional complicated manipulations of the integral. Such standard rules help simplify integration processes greatly, saving time during exams or homework.
To tackle trigonometric integrals efficiently, remember these tips:

• Memorize standard trigonometric integrals.
• Use trigonometric identities to simplify integrals when necessary.
• Practice different scenarios to become familiar with these patterns.

Differentiation

Differentiation is the process of finding the derivative of a function. It helps verify if you've calculated an antiderivative correctly.After finding an antiderivative, like \( \frac{2}{5} \sec \theta + C \) in this exercise, you take its derivative to check if you arrive back at the original function, \( \frac{2}{5} \sec \theta \tan \theta \).Let's break it down: - Differentiate \( \frac{2}{5} \sec \theta + C \). Using the derivative formula for \( \sec x \), which is \( \sec x \tan x \), we obtain \( \frac{2}{5} \sec \theta \tan \theta \) from \( \frac{2}{5} \sec \theta \).- The differentiation of "\(+ C\)," where \( C \) is a constant, is zero. This confirms that our solution is precise.Differentiation ensures the reliability of your integration results. Always check your integrals by differentiating the results.