Problem 35
Question
In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{\sec x+\cos x}{\cos x} $$
Step-by-Step Solution
Verified Answer
The general antiderivative is \( F(x) = \tan x + x + C \).
1Step 1: Simplify the Given Function
The first step is to simplify the expression \( f(x) = \frac{\sec x + \cos x}{\cos x} \). We know that \( \sec x = \frac{1}{\cos x} \). By substituting \( \sec x \) into the expression, we get: \[f(x) = \frac{\frac{1}{\cos x} + \cos x}{\cos x} = \frac{1}{\cos x \cdot \cos x} + \frac{\cos x}{\cos x} = \frac{1}{\cos^2 x} + 1\] We can rewrite \( \frac{1}{\cos^2 x} \) as \( \sec^2 x \), so the expression simplifies to:\[f(x) = \sec^2 x + 1\]
2Step 2: Find the Antiderivative
To find the antiderivative of \( f(x) = \sec^2 x + 1 \), consider the individual antiderivatives of each term. - The antiderivative of \( \sec^2 x \) is \( \tan x \) because \( \frac{d}{dx}(\tan x) = \sec^2 x \).- The antiderivative of 1 is \( x \) since \( \frac{d}{dx}(x) = 1 \).Thus, the general antiderivative of the function is:\[F(x) = \tan x + x + C\]where \( C \) is the constant of integration.
Key Concepts
Understanding AntiderivativesThe Role of Trigonometric FunctionsMastering Integration
Understanding Antiderivatives
An antiderivative, also known as an indefinite integral, is a fundamental concept in calculus. It is a function whose derivative is the original function you started with. If you have a function \( f(x) \), its antiderivative is a function \( F(x) \) such that \( F'(x) = f(x) \).
The process of finding an antiderivative is called integration, which is essentially reversing the process of differentiation. One notable point is that antiderivatives are not unique. They come with a constant of integration, \( C \), because when we differentiate a constant, we get zero, and therefore, we lose any information about a constant term when differentiating.
Therefore, when we write an antiderivative, we always add \( C \) to ensure we capture all possibilities that the antiderivative could encompass. In our example, where we found the antiderivative of \( f(x) = \sec^2 x + 1 \), the result is \( F(x) = \tan x + x + C \). This combination (including \( C \)) represents a family of functions whose derivative gives us back our original function \( f(x) \).
The process of finding an antiderivative is called integration, which is essentially reversing the process of differentiation. One notable point is that antiderivatives are not unique. They come with a constant of integration, \( C \), because when we differentiate a constant, we get zero, and therefore, we lose any information about a constant term when differentiating.
Therefore, when we write an antiderivative, we always add \( C \) to ensure we capture all possibilities that the antiderivative could encompass. In our example, where we found the antiderivative of \( f(x) = \sec^2 x + 1 \), the result is \( F(x) = \tan x + x + C \). This combination (including \( C \)) represents a family of functions whose derivative gives us back our original function \( f(x) \).
The Role of Trigonometric Functions
Trigonometric functions like \( \sec x \), \( \cos x \), and \( \tan x \) play a crucial role in calculus, especially when dealing with integrals and derivatives. These functions relate to the angles and sides of triangles, especially right-angled triangles, and they frequently appear in various integration problems.
In our exercise, the function given was \( f(x) = \frac{\sec x + \cos x}{\cos x} \). Simplifying using basic trigonometric identities, we rewrite it as \( \sec^2 x + 1 \). Recognizing these trigonometric functions is essential because many integration problems involve these identities.
It's important to memorize some common derivatives and antiderivatives of trigonometric functions:
In our exercise, the function given was \( f(x) = \frac{\sec x + \cos x}{\cos x} \). Simplifying using basic trigonometric identities, we rewrite it as \( \sec^2 x + 1 \). Recognizing these trigonometric functions is essential because many integration problems involve these identities.
It's important to memorize some common derivatives and antiderivatives of trigonometric functions:
- The derivative of \( \tan x \) is \( \sec^2 x \), so the antiderivative of \( \sec^2 x \) is \( \tan x \).
- The derivative of \( \sin x \) is \( \cos x \), and the antiderivative of \( \cos x \) is \( \sin x \), but with a "+C".
Mastering Integration
Integration is one of the two principal operations in calculus, the other being differentiation. While differentiation is concerned with finding the rate of change of a function, integration deals with finding the accumulated area under a curve or the total value of a function over an interval.
The integration process can be thought of as summing up small pieces to find a whole. There are two main types of integrals: definite and indefinite. In this context, we're focusing on indefinite integrals, which find antiderivatives and include the constant of integration \( C \).
In our problem, integration was used to find the antiderivative of \( \sec^2 x + 1 \). By knowing the basic antiderivatives—like that of \( \sec^2 x \) being \( \tan x \) and the antiderivative of 1 being \( x \)—we compiled these results to derive the general antiderivative, \( F(x) = \tan x + x + C \).
The integration process can be thought of as summing up small pieces to find a whole. There are two main types of integrals: definite and indefinite. In this context, we're focusing on indefinite integrals, which find antiderivatives and include the constant of integration \( C \).
In our problem, integration was used to find the antiderivative of \( \sec^2 x + 1 \). By knowing the basic antiderivatives—like that of \( \sec^2 x \) being \( \tan x \) and the antiderivative of 1 being \( x \)—we compiled these results to derive the general antiderivative, \( F(x) = \tan x + x + C \).
- Apply basic antiderivative formulas when solving integrals involving simple functions.
- Use algebraic manipulations or trigonometric identities to simplify complex integrands before integrating.
- Always include the constant \( C \) in indefinite integrals.
Other exercises in this chapter
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