Problem 55 - University Calculus: Early Transcendentals

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

Step 1: Identify the integral components

The given integral is \( \int \left( 4 \sec x \tan x - 2 \sec^2 x \right) \, dx \). We can split this into two separate integrals: \( 4 \int \sec x \tan x \, dx \) and \( -2 \int \sec^2 x \, dx \).

Step 2: Find the antiderivative of \( \sec x \tan x \)

Recall that the derivative of \( \sec x \) is \( \sec x \tan x \). Therefore, the antiderivative of \( \sec x \tan x \) is \( \sec x \).

Step 3: Find the antiderivative of \( \sec^2 x \)

Recall that the derivative of \( \tan x \) is \( \sec^2 x \). Thus, the antiderivative of \( \sec^2 x \) is \( \tan x \).

Step 4: Combine results with given coefficients

Using the antiderivatives obtained, we have: \( 4 \int \sec x \tan x \, dx = 4 \sec x + C_1 \) and \( -2 \int \sec^2 x \, dx = -2 \tan x + C_2 \). Combine these results to get: \( 4 \sec x - 2 \tan x + C \), where \( C = C_1 + C_2 \).

Step 5: Verify by differentiation

Differentiate \( 4 \sec x - 2 \tan x + C \): the derivative of \( 4 \sec x \) is \( 4 \sec x \tan x \) and the derivative of \( -2 \tan x \) is \( -2 \sec^2 x \). Therefore, the derivative of \( 4 \sec x - 2 \tan x + C \) is indeed \( 4 \sec x \tan x - 2 \sec^2 x \), confirming our solution.

Key Concepts

Understanding Antiderivatives

An antiderivative, sometimes also called an indefinite integral, is a function that reverses differentiation. Essentially, if you differentiate a function, and then integrate the resulting expression, you should return to your original function (plus a constant). This constant, often denoted as "C", represents any constant term that was lost during differentiation.
This is because differentiation eliminates constants, showing why antiderivatives are not unique. Consider how the task looks like with basic functions:

• If the derivative of a function is 2, the antiderivative could be 2x or 2x + 5, or even 2x + 10.
• All of these functions, once differentiated, return 2, since the derivative of any constant is 0.

In practice, when asked to find the indefinite integral, identify the main function to integrate and remember to add the constant of integration at the end. This ensures that your answer includes all possible original functions.

Integration Techniques

Integration techniques are methods to find the integral of a function. In the given problem, we use direct integration which applies when the antiderivative is recognized through known derivatives of basic functions or components. This might sound complicated, but it really boils down to recognizing patterns.When solving the original exercise, the integral is split into two parts:

• Integrating \( 4 \sec x \tan x \)
• Integrating \( -2 \sec^2 x \)

It’s very effective to use this method of splitting because it simplifies the task. You tackle each part individually, relying on basic antiderivatives. This technique is based on recognizing that \( \sec x \tan x \) is the derivative of \( \sec x \), and \( \sec^2 x \) is the derivative of \( \tan x \).
Such methods improve accuracy when searching for an antiderivative. Always consider whether you can simplify an integral by breaking it down into recognizable components, as was done in the solution. The better you are at spotting these patterns, the easier integration will become.

Trigonometric Functions and Their Antiderivatives

Trigonometric functions play a crucial role in calculus, and it helps to memorize some basic antiderivatives. Recognizing these can save a lot of time:

• The antiderivative of \( \sec x \tan x \) is \( \sec x \), because when you differentiate \( \sec x \), you return to \( \sec x \tan x \).
• Similarly, the antiderivative of \( \sec^2 x \) is \( \tan x \), as differentiating \( \tan x \) yields \( \sec^2 x \).

These relationships are part of what makes calculus powerful. Once these are understood and memorized, solving such integrals becomes a matter of applying known rules and techniques.
Trigonometric functions often appear in a variety of integrals. Understanding their derivatives helps in finding antiderivatives efficiently. This understanding simplifies even more complex integration tasks in calculus.