Problem 42 - 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. $$\int \frac{2}{5} \sec \theta \tan \theta d \theta$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral

We need to find the indefinite integral of \( \frac{2}{5} \sec \theta \tan \theta \). This integral resembles a common trigonometric integral form.

Step 2: Recognize Standard Form

We notice that the integral \( \int \sec \theta \tan \theta \ d\theta \) is a standard form, and its antiderivative is \( \sec \theta \).

Step 3: Apply Linearity of Integration

Using the property of linearity in integration, we can take the constant \( \frac{2}{5} \) outside the integral. So, we rewrite it as \( \frac{2}{5} \int \sec \theta \tan \theta \ d\theta \).

Step 4: Find the Antiderivative

Based on the standard form recognition, we can write the antiderivative as \( \frac{2}{5} \sec \theta + C \) where \( C \) is the constant of integration.

Step 5: Verify by Differentiation

Differentiate \( \frac{2}{5} \sec \theta + C \) with respect to \( \theta \). The derivative of \( \sec \theta \) is \( \sec \theta \tan \theta \), and \( C \) differentiates to 0. Thus, the derivative is \( \frac{2}{5} \sec \theta \tan \theta \), which matches the original integrand.

Key Concepts

Antiderivative

The concept of an antiderivative is essential when dealing with indefinite integrals. An antiderivative is a function that, when differentiated, results in the original function or integrand. It's like reversing the process of differentiation.

In simple terms: if differentiation takes a function and finds its rate of change, integration does the opposite, recovering the original function from its rate of change.

For an indefinite integral, the aim is to find all possible functions (antiderivatives) that can differentiate to give the original integrand.

• The result of an indefinite integral is a family of functions, expressed with a constant of integration, denoted as "C".
• This constant represents any constant value that could be added to the function, since the derivative of a constant is zero.
• In the example given, the antiderivative of \( \int \frac{2}{5} \sec \theta \tan \theta \, d\theta \) is \( \frac{2}{5} \sec \theta + C \).

Identifying the antiderivative is crucial for understanding indefinite integrals. Make sure to learn the common forms of antiderivatives, especially for basic trigonometric functions.

Trigonometric Integrals

Trigonometric integrals are an interesting class of integrals involving trigonometric functions like sine, cosine, tangent, secant, and their reciprocals. Recognizing standard forms of trigonometric integrals can greatly simplify the integration process.

In trigonometry, some functions appear frequently, such as \( \int \sec \theta \tan \theta \, d\theta \), which is common due to its direct antiderivative, \( \sec \theta \).

• It's useful to remember that differentiating \( \sec \theta \) yields \( \sec \theta \tan \theta \), which means integrating \( \sec \theta \tan \theta \) gives us \( \sec \theta \).
• Understanding these standard forms can aid in quickly identifying solutions to problems involving trigonometric integrals.
• Often, these integrals might appear complex, but recognizing their standard forms can make solving them straightforward.

When dealing with trigonometric integrals, it's helpful to practice recognizing typical patterns to simplify your work.

Linearity of Integration

Linearity of integration is a fundamental property that allows us to simplify the process of finding integrals. This property tells us two very important things:

Firstly, we can separate sums into individual integrals. Secondly, constants can be factored out from integrals. This means that if you have an integral like \( a \cdot f(x) \), you can rewrite it as \( a \int f(x) \, dx \).

Applying linearity:

• So when approaching \( \int \frac{2}{5} \sec \theta \tan \theta \, d\theta \), we can move the constant \( \frac{2}{5} \) outside to get \( \frac{2}{5} \int \sec \theta \tan \theta \, d\theta \).
• This makes the integration process easier since we only need to focus on finding the integral of the trigonometric part, \( \int \sec \theta \tan \theta \, d\theta \).

Linearity is a powerful tool for simplifying complex integrals. Utilizing this can provide a clearer path to solving various indefinite integrals. Practicing applying the principles of linearity will make your integration tasks much smoother.