/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Why is this section called 鈥淭h... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 77

Why is this section called 鈥淭he Indefinite Integral?鈥�

Short Answer

Expert verified
The section is called "The Indefinite Integral" due to the presence of an arbitrary constant, denoted by "C", in the result of the integration. Since the constant "C" can take any real value, this makes the solution indefinite, implying that there might be multiple solutions. An indefinite integral represents a family of functions with all possible values of "C", contrasting with definite integrals that yield a specific numerical value.

Step by step solution

01

Define the Indefinite Integral

An indefinite integral, denoted by the symbol \(\int\), is one of the two main types of integrals in calculus (the other being definite integral). It represents the anti-derivative of a function, which means that it is the reverse process of differentiation. The symbol \(\int\) represents the integration process, and an indefinite integral is written in the form: \[ \int f(x) dx \] Where \(f(x)\) represents the function to be integrated, and \(dx\) shows that we are integrating with respect to the variable \(x\).
02

Distinct Features of Indefinite Integrals

An essential feature of indefinite integrals is that the result is a function rather than a single value. This reveals the fact that there might be more than one anti-derivative or solution for a given function. Another important characteristic of indefinite integrals is that they involve an arbitrary constant denoted by "C". This happens because differentiating a constant results in zero, and when we take the anti-derivative, we need to account for this constant. Thus, the result of an indefinite integral is written as: \[ \int f(x) dx = F(x) + C \] Where \(F(x)\) is the anti-derivative of \(f(x)\), and \(C\) represents the arbitrary constant.
03

Why is it called "The Indefinite Integral"?

The term "indefinite" refers to the arbitrariness of the constant in the result of the integration. As the constant "C" is unknown and can take any real value, it makes the solution "indefinite", that is, there might be more than one solution. Furthermore, an indefinite integral's result represents a family of functions with all possible values of "C", rather than a single function or specific value. This stands in contrast with definite integrals, which exhibit a definite result, a single numerical value representing the signed area between the function and the \(x\)-axis. In conclusion, the section is called "The Indefinite Integral" because the integral in question does not provide an exact solution but instead yields a family of functions with an arbitrary constant, making the result indefinite.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Anti-Derivative
The concept of an anti-derivative is fundamental in the study of calculus. It involves finding a function whose derivative is the given function. In simple terms, if you have a function, the anti-derivative is a function that, when differentiated, returns the original function.

For instance, if you know that the derivative of a function is a polynomial like \(f(x) = 2x\), finding its anti-derivative means you want to find a function whose derivative is \(2x\). The answer would be \(F(x) = x^2 + C\), where \(C\) is the arbitrary constant. This is because when you differentiate \(F(x) = x^2 + C\), you get back to \(2x\), since the derivative of a constant is zero.

This process of finding the anti-derivative is what we refer to as integration, specifically indefinite integration. By reversing the process of differentiation, it emphasizes the inverse relationship between derivatives and integrals.
Integration
Integration, in the context of indefinite integrals, refers to the process of finding the anti-derivative of a function. Denoted by the integral sign \(\int\), integration is the inverse process of differentiation. It serves to "undo" the derivative and return to the original function.

The expression \(\int f(x) \, dx\) represents the indefinite integral of \(f(x)\) with respect to \(x\). One key point about this type of integration is that it results in a family of functions rather than a single value. This is because the anti-derivative is not unique unless initial conditions are provided.

Through integration, we collect all the possible functions whose derivative would result in the given function \(f(x)\). This is contrasted with definite integrals, which compute a specific numerical value representing area under a curve.
  • Reverse Process: Integration is essentially finding the anti-derivative.
  • Notation: The integral symbol \(\int\) signals integration is taking place.
  • Result: Produces a family of functions, often expressed with an arbitrary constant.
Arbitrary Constant
When finding the indefinite integral of a function, the solution is not complete without the arbitrary constant, often denoted as \(C\). This constant represents any fixed value that could have been added to the function before differentiation, as the derivative of any constant is zero.

Consider the integral \(\int 2x \, dx\). Performing the integration, we arrive at \(x^2 + C\). The addition of \(C\) is crucial because there are infinitely many solutions, each differing by a constant amount. Thus, the presence of \(C\) indicates this infinity of possibilities, representing a "family of functions."

The arbitrary constant serves as a reminder of the general nature of anti-derivatives. It indicates that any specific value for \(C\) "shifts" the basic anti-derivative function vertically. Hence, without additional information (like initial conditions), \(C\) remains undetermined, embodying the "indefinite" aspect of indefinite integrals.
  • Nature: \(C\) represents an infinite number of function shifts.
  • Importance: Ensures completeness of the integral.
  • Indication: Demonstrates the indefinite potential solutions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

a. Show that the logistic function \(f(x)=\frac{N}{1+A e^{-k x}}\) can be written in the form $$ f(x)=\frac{N e^{k x}}{A+e^{k x}} $$ HINT [See the note after Example 7 in Section 13.2.] b. Use the result of part (a) and a suitable substitution to show that $$ \int \frac{N}{1+A e^{-k x}} d x=\frac{N \ln \left(A+e^{k x}\right)}{k}+C $$ c. The rate of spending on grants by U.S. foundations in the period 1993-2003 was approximately \(s(t)=11+\frac{20}{1+1,800 e^{-0.9 t}}\) billion dollars per year $$ (3 \leq t \leq 13) $$ where \(t\) is the number of years since \(1990 .^{52}\) Use the result of part (b) to estimate, to the nearest \(\$ 10\) billion, the total spending on grants from 1998 to 2003 .

Evaluate the integrals. $$ \int_{-\sqrt{2}}^{\sqrt{2}} 3 x \sqrt{2 x^{2}+1} d x $$

(Compare Exercise 41 in Section 13.3.) The rate of U.S. sales of bottled water for the period \(2000-2008\) can be approximated by \(s(t)=12 t^{2}+500 t+4,700\) million gallons per year $$ (0 \leq t \leq 8) $$ where \(t\) is time in years since the start of \(2000 .{ }^{37}\) Use the FTC to estimate the total U.S. sales of bottled water from the start of 2000 to the start of 2005 . (Round your answer to the nearest billion gallons.)

Evaluate the integrals. $$ \int_{0}^{1} 8(-x+1)^{7} d x $$

Calculate the total area of the regions described. Do not count area beneath the \(x\) -axis as negative. HINT [See Example 6.] Bounded by the \(x\) -axis, the curve \(y=x e^{x^{2}-1}\), and the lines \(x=0\) and \(x=1\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.