Problem 43 - 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. $$\int\left(4 \sec x \tan x-2 \sec ^{2} x\right) d x$$

Step-by-Step Solution

Answer

\(4\sec x - 2\tan x + C\)

Step 1: Recognize the basic integrals

Recall that the derivative of \( \sec x \) is \( \sec x \tan x \) and the derivative of \( \tan x \) is \( \sec^2 x \). This helps in recognizing what the potential antiderivative functions could be.

Step 2: Break down the integral

The given integral is \( \int (4 \sec x \tan x - 2 \sec^2 x) \, dx \), which can be separated as two integrals: \( \int 4 \sec x \tan x \, dx - \int 2 \sec^2 x \, dx \).

Step 3: Find the antiderivative of the first term

The integral \( \int 4 \sec x \tan x \, dx \) is solved as \( 4 \int \sec x \tan x \, dx \). From Step 1, we know that \( \int \sec x \tan x \, dx = \sec x + C \), so \( 4 \int \sec x \tan x \, dx = 4\sec x + C_1 \).

Step 4: Find the antiderivative of the second term

The second integral \( \int 2 \sec^2 x \, dx \) simplifies to \( 2 \int \sec^2 x \, dx \). Knowing that \( \int \sec^2 x \, dx = \tan x + C \), it becomes \( 2\tan x + C_2 \).

Step 5: Combine the results

Combine the antiderivatives obtained from Steps 3 and 4: \( 4\sec x - 2\tan x + C \). The \( C \) is a constant representing \( C_1 - C_2 \).

Step 6: Verify the solution by differentiation

Differentiate \( 4\sec x - 2\tan x + C \) to check if it gives the original integrand. The derivative of \( 4\sec x \) is \( 4\sec x \tan x \) and the derivative of \( -2\tan x \) is \(-2 \sec^2 x \). Hence, the derivative is \( 4\sec x \tan x - 2\sec^2 x \), matching the integrand.

Key Concepts

Indefinite Integrals

An indefinite integral, also known as an antiderivative, is a fundamental concept in calculus that helps us find a function whose derivative is given. Unlike definite integrals, which calculate area under a curve, indefinite integrals express a family of functions. These functions differ by a constant, often noted as "+ C" because the process of differentiation loses constant values.

When working with indefinite integrals, particularly those involving trigonometric functions, it is crucial to recognize standard integral formulas. For example:

• \( \int \sec x \tan x \, dx = \sec x + C \)
• \( \int \sec^2 x \, dx = \tan x + C \)

These forms help in quickly identifying and calculating antiderivatives, allowing you to break down complex expressions into simpler parts. Always remember that each indefinite integral needs the + C to account for any constant displacement inherent in integration.

Trigonometric Functions

Trigonometric functions such as \( \sec x \), \( \tan x \), and related identities are vital in solving calculus problems, particularly indefinite integrals. Understanding the derivatives of these functions is essential as it connects to their integrals.

Key derivatives include:

• \( \frac{d}{dx}(\sec x) = \sec x \tan x \)
• \( \frac{d}{dx}(\tan x) = \sec^2 x \)

These derivatives are the building blocks for solving integrals. For instance, when you encounter \( \int 4 \sec x \tan x - 2 \sec^2 x \, dx \), recognizing that \( \sec x \tan x \) derives from \( \sec x \) simplifies the integration process.

Moreover, familiarity with trigonometric identities can help simplify and solve integrals. Utilizing relationships and identities can transform a challenging integral into a more manageable form.

Differentiation

Differentiation is the process of finding the derivative, or the rate of change, of a function. When verifying solutions to indefinite integrals, differentiating the resulting antiderivative is an essential step to ensure accuracy.

Differentiation involves applying rules such as the product and chain rules to determine how a function changes at any given point. In the context of verifying integration results:

• The derivative of \( 4\sec x \) is \( 4\sec x \tan x \).
• The derivative of \(-2\tan x\) is \(-2 \sec^2 x \).

Combining these results, if the derivative matches the original integrand, it confirms that the integration was conducted correctly.

This verification step not only solidifies understanding of differentiation but also strengthens problem-solving skills in calculus. It demonstrates the interconnected nature of integration and differentiation, reinforcing the concepts of calculus as a cohesive discipline.