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

Step-by-Step Solution

Answer

The most general antiderivative is \( 4 \sec x - 2 \tan x + C \).

Step 1: Identify Integral Components

Notice that the integral is of the form \( \int (4 \sec x \tan x - 2 \sec^2 x) \, dx \). This suggests splitting the integral into separate parts: \( \int 4 \sec x \tan x \, dx \) and \( \int -2 \sec^2 x \, dx \).

Step 2: Integrate Each Component Separately

Integrate the first part \( \int 4 \sec x \tan x \, dx \). Recall that the derivative of \( \sec x \) is \( \sec x \tan x \). Hence, the integral is \( 4 \sec x + C_1 \).

Step 3: Integrate the Second Component

Integrate the second part \( \int -2 \sec^2 x \, dx \). Recall that the derivative of \( \tan x \) is \( \sec^2 x \). Hence, the integral is \( -2 \tan x + C_2 \).

Step 4: Combine the Integrals

Combine the results from Steps 2 and 3. The most general antiderivative is \( 4 \sec x - 2 \tan x + C \), where \( C = C_1 + C_2 \) is a constant of integration.

Step 5: Verify by Differentiation

Differentiate the result \( 4 \sec x - 2 \tan x + C \). The derivative of \( 4 \sec x \) is \( 4 \sec x \tan x \), and the derivative of \( -2 \tan x \) is \( -2 \sec^2 x \). Thus, the derivative is \( 4 \sec x \tan x - 2 \sec^2 x \), which matches the original integrand, confirming the solution.

Key Concepts

Indefinite Integral

An indefinite integral, often referred to simply as an "integral," is a fundamental concept in calculus. It's the process through which one finds the antiderivative of a function. Unlike definite integrals, indefinite integrals do not have set limits. Instead, they represent a family of functions which differ by a constant. This is because when you take the derivative of a constant, it becomes zero, so the original function could have had any constant added to it.
In our given problem, the indefinite integral to find is:

• \( \int \left( 4 \sec x \tan x - 2 \sec^2 x \right) \, dx \)

The goal is to determine a function whose derivative results in the integrand, which is inside the integral sign. By solving for this integral, we represent the indefinite integral with an expression plus an arbitrary constant \( C \). The inclusion of this constant is crucial, as it accounts for all possible antiderivatives of the original function.

Antiderivative

An antiderivative is essentially the reverse process of differentiation. It involves finding a function, from the family of functions, whose derivative is equal to the given function. In terms of notation, if \( F(x) \) is an antiderivative of \( f(x) \), then:

• \( \frac{d}{dx}[F(x)] = f(x) \)

In the solution steps provided, the task was to find the antiderivative of the integrand \( 4 \sec x \tan x - 2 \sec^2 x \). By identifying components of the integral and integrating separately:

• For \( 4 \sec x \tan x \), the antiderivative becomes \( 4 \sec x \).
• For \( -2 \sec^2 x \), the antiderivative becomes \( -2 \tan x \).

Combining these results gives the most general antiderivative:

• \( 4 \sec x - 2 \tan x + C \)

This expression includes a constant \( C \), signifying the infinite number of solutions that differ only by a constant.

Verification by Differentiation

Verification by differentiation is the final check to ensure the antiderivative found is correct. It's a straightforward process:

• You compute the derivative of the antiderivative expression found from integration.
• If the resulting derivative matches the original function to be integrated, your solution is verified.

In this exercise, the antiderivative expression derived was \( 4 \sec x - 2 \tan x + C \). By differentiating this, we perform the following steps:

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

Together, these yield the original integrand \( 4 \sec x \tan x - 2 \sec^2 x \).
Since the differentiation of the antiderivative led back to the given integrand, this confirms the correctness of the solution. This step is critical in calculus as it anchors the correctness of indefinite integrals through differentiation.