Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Antiderivatives

Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

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2:30 minutes

Problem 4.9.3

Textbook Question

Describe the set of antiderivatives of ƒ(x) = 1

Verified step by step guidance

Recognize that the antiderivative of a function ƒ(x) is a function F(x) such that the derivative of F(x) equals ƒ(x). In this case, we are given ƒ(x) = 1.

Recall the basic rule of integration: the antiderivative of a constant c is c * x + C, where C is the constant of integration. Here, the constant c is 1.

Apply the rule to find the antiderivative of ƒ(x) = 1. The result is F(x) = x + C, where C represents an arbitrary constant.

Understand that the set of antiderivatives of ƒ(x) = 1 is represented by the family of functions F(x) = x + C, where C can be any real number.

Conclude that the set of antiderivatives is infinite, as it depends on the value of the constant C, which can vary freely.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Antiderivative

An antiderivative of a function f(x) is another function F(x) such that the derivative of F(x) equals f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x). Antiderivatives are essential in calculus as they are used to find the area under curves and solve differential equations.

Constant Functions

The function f(x) = 1 is a constant function, meaning it has the same value (1) for all x in its domain. The antiderivative of a constant function is a linear function, which can be expressed as F(x) = x + C, where C is a constant. This reflects the fact that the slope of the constant function is zero, leading to a linear increase in the antiderivative.

Family of Antiderivatives

The set of all antiderivatives of a function forms a family of functions that differ only by a constant. For the function f(x) = 1, the family of antiderivatives can be expressed as F(x) = x + C, where C represents any real number. This indicates that there are infinitely many antiderivatives, each corresponding to a different vertical shift of the linear function.

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Multiple Choice

Let $h (x)$ be an antiderivative of the function $f (x) = x^{3} + \frac{- \sin x}{x^{2}} + 2$ . Which of the following is a correct expression for $h (x)$ ?