Problem 52 - Thomas Calculus

Question

In Exercises \(17-56,\) 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. $$\int\left(2+\tan ^{2} \theta\right) d \theta$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Integral Expression

To find the indefinite integral \( \int (2 + \tan^2 \theta) \, d\theta \), notice that we can rewrite the integrand using a trigonometric identity:\[ \tan^2 \theta = \sec^2 \theta - 1 \]This gives:\[ \int (2 + \tan^2 \theta) \, d\theta = \int (2 + \sec^2 \theta - 1) \, d\theta = \int (1 + \sec^2 \theta) \, d\theta \]

Step 2: Perform the Integration

Now integrate each term of the expression:- The integral of \(1\) with respect to \(\theta\) is simply \( \theta \).- The integral of \(\sec^2 \theta\) with respect to \(\theta\) is \( \tan \theta \).Thus, the antiderivative is:\[ \int (1 + \sec^2 \theta) \, d\theta = \theta + \tan \theta + C \]where \(C\) is the constant of integration.

Step 3: Verify by Differentiation

Differentiate the result \( \theta + \tan \theta + C \) to verify it is correct:- The derivative of \( \theta \) is \( 1 \).- The derivative of \( \tan \theta \) is \( \sec^2 \theta \).Therefore, the derivative is:\[ 1 + \sec^2 \theta \]This matches the integrand from the rewritten integral, confirming our solution is correct.

Key Concepts

Antiderivative

The concept of antiderivatives is central to understanding indefinite integrals. An antiderivative of a function is a function whose derivative yields the original function. In simpler terms, it's the reverse process of differentiation.
Think of it as finding a function that represents the accumulated area under the curve of another function.When working with indefinite integrals, our goal is to determine an antiderivative. This process has no bounds, thus the solution includes a constant of integration, noted as **C**. This constant reflects any vertical shift in the antiderivative's graph.For instance, when finding the antiderivative of 1 with respect to \( \theta \), we simply obtain \( \theta \). Similarly, integrating \( \sec^2 \theta \) gives us \( \tan \theta \). The final antiderivative is then expressed as \( \theta + \tan \theta + C \).

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domain. These identities are extremely useful in simplifying integrals involving trigonometric functions.In the given exercise, the identity \( \tan^2 \theta = \sec^2 \theta - 1 \) helps transform the original integral into a simpler form. By substituting this identity into \( \int (2 + \tan^2 \theta) \, d\theta \), we obtain \( \int (1 + \sec^2 \theta) \, d\theta \).This transformation process can make complex integrals more manageable by turning them into a form that is easier to integrate directly.
These simplifications are essential strategies in calculus, particularly for integrating functions that initially appear intricate.

Trigonometric Integrals

Trigonometric integrals are integrals involving trigonometric functions like sine, cosine, tangent, and secant. They often require special techniques and identities to solve.In this exercise, after applying the trigonometric identity, we have the simpler integral \( \int (1 + \sec^2 \theta) \, d\theta \). Integrating each part separately, we find:

• \( \int 1 \, d\theta = \theta \)
• \( \int \sec^2 \theta \, d\theta = \tan \theta \)

Combining these results gives us the antiderivative \( \theta + \tan \theta + C \).Recognizing and using trigonometric integrals is an essential skill in calculus. It involves both understanding the properties of trigonometric functions and applying integration techniques to solve a variety of problems.