Problem 49 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \begin{array}{l}{\int\left(1+\tan ^{2} \theta\right) d \theta} \\\ {\left(\text {Hint} : 1+\tan ^{2} \theta=\sec ^{2} \theta\right)}\end{array} $$

Step-by-Step Solution

Answer

The antiderivative is \( \tan \theta + C \).

Step 1: Identify the Integral

The given integral is \( \int (1 + \tan^2 \theta) \, d\theta \). The problem provides a hint that \( 1 + \tan^2 \theta = \sec^2 \theta \). This means we can rewrite the integral as \( \int \sec^2 \theta \, d\theta \).

Step 2: Integrate the Function

The integral of \( \sec^2 \theta \) is a well-known result. We have \( \int \sec^2 \theta \, d\theta = \tan \theta + C \), where \( C \) is the constant of integration. Hence, the most general antiderivative is \( \tan \theta + C \).

Step 3: Differentiate to Verify

To verify our result, differentiate the function \( \tan \theta + C \). The derivative of \( \tan \theta \) is \( \sec^2 \theta \), and the derivative of the constant \( C \) is 0. Thus, the derivative of \( \tan \theta + C \) is \( \sec^2 \theta \), which matches the integrand we started with, confirming our solution is correct.

Key Concepts

Indefinite Integral

An indefinite integral, often called an antiderivative, is a function that reverses the process of differentiation. When you find the indefinite integral of a function, you're essentially looking for a function whose derivative is the original function you started with. In mathematical terms, the indefinite integral of a function \( f(x) \) with respect to \( x \) is given by \( \int f(x) \, dx = F(x) + C \), where \( F(x) \) is the antiderivative of \( f(x) \) and \( C \) is the constant of integration. This constant \( C \) represents any constant value, since the derivative of any constant is zero.
Understanding indefinite integrals is crucial, as they form the basis for solving various mathematical and real-world problems. The integral notation \( \int \) can be seen as the opposite of differentiation, providing a broader picture of the function's behavior.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved. They are fundamental tools in calculus, especially when dealing with integrals and derivatives of trigonometric functions.

• The Pythagorean identity, \( 1 + \tan^2 \theta = \sec^2 \theta \), was used in the original exercise to simplify the integrand. By recognizing this identity, the integral \( \int(1 + \tan^2 \theta) \, d\theta \) was transformed into a simpler form \( \int \sec^2 \theta \, d\theta \).
• Using these identities effectively can simplify complex integrals and make the calculation of antiderivatives more straightforward.

Recognizing and applying these identities is a skill that will help solve a wide range of trigonometric problems with ease.

Differentiation Verification

Once you've found an antiderivative, it's important to verify your solution. Differentiation verification involves taking the derivative of your antiderivative to see if it yields the original function you started with.
For example, in the solution \( \tan \theta + C \), where \( C \) is a constant, you differentiate \( \tan \theta \) to see if you return to \( \sec^2 \theta \). The derivative of \( \tan \theta \) is \( \sec^2 \theta \), and the derivative of a constant is 0. Therefore, differentiating \( \tan \theta + C \) gives \( \sec^2 \theta \), which is the original integrand in the exercise. This confirms the correctness of the integration process.

• This process not only checks your work but also reinforces your understanding of the relationship between differentiation and integration.

Using differentiation to verify antiderivatives is a reliable method to ensure the accuracy of the solutions in calculus.