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Chapter 4: Problem 43

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

Short Answer

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

Step by step solution

01

Identify the Integral to Solve

We are given the integral \( \int \left( 4 \sec x \tan x - 2 \sec^2 x \right) \, dx \). Our goal is to find the antiderivative of this expression.
02

Separate the Integral

Separate the integral into two separate integrals:\[\int 4 \sec x \tan x \, dx - \int 2 \sec^2 x \, dx\]
03

Recall Basic Antiderivatives

Recall that the antiderivative of \( \sec x \tan x \) is \( \sec x \), and the antiderivative of \( \sec^2 x \) is \( \tan x \).
04

Solve the First Integral

The first integral is \( \int 4 \sec x \tan x \, dx \). Using the knowledge from Step 3, we have:\[4 \int \sec x \tan x \, dx = 4 \sec x + C_1\]
05

Solve the Second Integral

The second integral is \( \int 2 \sec^2 x \, dx \). Using the antiderivative from Step 3, we find:\[2 \int \sec^2 x \, dx = 2 \tan x + C_2\]
06

Combine and Simplify the Solution

Combine the adjusted results from Steps 4 and 5:\[4 \sec x - 2 \tan x + C\]Here, \( C = C_1 - C_2 \) represents the constant of integration.
07

Verify the Solution by Differentiation

Differentiate \( 4 \sec x - 2 \tan x + C \) with respect to \( x \):1. The derivative of \( 4 \sec x \) is \( 4 \sec x \tan x \).2. The derivative of \( -2 \tan x \) is \( -2 \sec^2 x \).3. The derivative of the constant \( C \) is 0.So, the derivative of the antiderivative is \( 4 \sec x \tan x - 2 \sec^2 x \), matching the original integrand.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
An antiderivative, also known as an indefinite integral, is a function that reverses differentiation. If you have the derivative of a function and want to find the original function, you're looking for its antiderivative. In the context of the integral given in the exercise, the antiderivative is a function whose derivative yields the integrand.

The key properties to remember about antiderivatives include:
  • The antiderivative of a sum or difference of functions is the sum or difference of their antiderivatives.
  • The antiderivative is not unique; it includes a constant of integration, denoted by \( C \).
  • Finding an antiderivative is essentially the reverse process of differentiation.
Using these properties, the integral \( \int(4 \sec x \tan x - 2 \sec^2 x) \, dx \) was evaluated by separating it into simpler parts and solving each separately.
Integration Techniques
Various integration techniques can help solve complex integrals effectively. The most common methods include basic antiderivative rules, substitution, integration by parts, and partial fraction decomposition.

For the given integral, we used basic antiderivative rules where known forms of trigonometric derivatives were applied:
  • Recognize that the antiderivative of \( \sec x \tan x \) is \( \sec x \).
  • The antiderivative of \( \sec^2 x \) is \( \tan x \).
These observations helped directly find the antiderivative of each part of the integral. Because the integral was expressed as a difference, each component could be solved independently, simplifying the calculation process significantly. Always remember to add a constant \( C \) once you've found the antiderivative, since integration involves an indefinite process leading to a family of curves.
Trigonometric Integrals
Trigonometric integrals involve the integration of functions containing trigonometric functions like \( \sin x \), \( \cos x \), \( \tan x \), and \( \sec x \). To tackle these integrals, recalling the derivatives and antiderivatives of standard trigonometric functions is crucial.

For example, in our exercise:
  • The derivative of \( \sec x \) is \( \sec x \tan x \), helping to determine the antiderivative when reversed.
  • The derivative of \( \tan x \) is \( \sec^2 x \), used similarly for identifying antiderivatives.
Knowing these relationships allows the integration process to be more straightforward. Moreover, because trigonometric functions often appear in calculus, being familiar with these basic derivatives and antiderivatives is particularly helpful.

Always check your work by differentiating your antiderivative if possible. As shown in the solution, differentiating the antiderivative \( 4 \sec x - 2 \tan x + C \) confirmed it matched the original function sought by the integral, proving the solution was correct.

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