Problem 42 - Thomas Calculus

Question

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

The integrand is \( \frac{2}{5} \sec \theta \tan \theta \). Your goal is to find the antiderivative of this function.

Step 2: Identify Basic Antiderivative

The function \( \sec \theta \tan \theta \) is a known derivative. Specifically, the derivative of \( \sec \theta \) is \( \sec \theta \tan \theta \). This means that the antiderivative of \( \sec \theta \tan \theta \) is \( \sec \theta \).

Step 3: Apply the Constant Multiple in Antiderivative

Since our function is \( \frac{2}{5} \sec \theta \tan \theta \), we take the constant \( \frac{2}{5} \) outside the integral sign. Thus, the antiderivative is \( \frac{2}{5} \sec \theta \).

Step 4: Add Constant of Integration

The indefinite integral includes a constant of integration. Therefore, the most general antiderivative is \( \frac{2}{5} \sec \theta + C \), where \( C \) is the constant of integration.

Step 5: Differentiate to Check the Answer

Differentiate \( \frac{2}{5} \sec \theta + C \) to confirm the solution is correct. By the differentiation rule, the derivative of \( \sec \theta \) is \( \sec \theta \tan \theta \). Hence, the derivative of \( \frac{2}{5} \sec \theta \) is \( \frac{2}{5} \sec \theta \tan \theta \), which matches the original integrand.

Key Concepts

Antiderivative

An antiderivative is a function that reverses the process of differentiation. In other words, if you take the derivative of a function and then find its antiderivative, you should get back the original function. This concept is crucial when dealing with indefinite integrals. When you see the integral symbol, \( \int \), it signifies the process of finding an antiderivative. This process is like working backward from the derivative.
The given problem involves finding the antiderivative of \( \frac{2}{5} \sec \theta \tan \theta \). A key thing to notice is that \( \sec \theta \tan \theta \) is a known derivative. Specifically, it's the derivative of \( \sec \theta \). This means that the antiderivative of \( \sec \theta \tan \theta \) is simply \( \sec \theta \).

• You recognize this relationship helps you solve the integral more efficiently.
• Understanding common derivatives and their antiderivatives is crucial.

By taking constants out of the integration process, as seen in the exercise, you simplify the integration to mainly focus on the function itself.

Constant of Integration

Whenever you find an indefinite integral, or antiderivative, a constant of integration, denoted by \( C \), is included in your answer. This constant is critical because differentiation of a constant is zero, and without it, you'd miss potential solutions. The inclusion of \( C \) represents all the possible vertical shifts of the antiderivative graph on the coordinate plane.
The concept comes into play when considering that there are infinitely many antiderivatives for any given function. By adding \( C \), you capture each of these possibilities.

• The constant of integration is what makes an antiderivative general.
• Without adding \( C \), your solution is considered incomplete.

In our example, the most general form of the antiderivative is \( \frac{2}{5} \sec \theta + C \), ensuring all solutions to the indefinite integral are included.

Trigonometric Functions

Trigonometric functions like \( \sec \theta \) and \( \tan \theta \) play a vital role in calculus, particularly when dealing with integrals. These functions are not only significant on their own but also in combinations, such as the given \( \sec \theta \tan \theta \). Each trigonometric function has its derivatives and antiderivatives, which are often used in integration.
For example, the derivative of \( \sec \theta \) is \( \sec \theta \tan \theta \), which is why, in our exercise, knowing this helps find its antiderivative swiftly. Mastery of trigonometric identities and derivatives is helpful, especially when simplifying integrals.

• Identifying these patterns makes integration much smoother.
• Trigonometric derivatives are foundational rules you'll use frequently.

Understanding these functions and their properties is key to solving integrals involving trigonometric elements effectively.