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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2:53 minutes

Problem 4.9.7

Textbook Question

Give the antiderivatives of 1/x.

Verified step by step guidance

Recognize that the function 1/x is a standard form for which the antiderivative is well-known. The antiderivative of 1/x is related to the natural logarithm function.

Recall the rule for the antiderivative of 1/x: ∫(1/x) dx = ln|x| + C, where ln|x| represents the natural logarithm of the absolute value of x, and C is the constant of integration.

Understand why the absolute value is necessary: The natural logarithm function, ln(x), is only defined for positive values of x. To account for both positive and negative values of x, we use ln|x|.

Write the general form of the antiderivative: ∫(1/x) dx = ln|x| + C. This represents the family of functions whose derivative is 1/x.

Conclude that the antiderivative of 1/x is ln|x| + C, emphasizing the importance of including the constant of integration (C) to represent all possible antiderivatives.

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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 is another function whose derivative is the original function. In calculus, finding the antiderivative is a fundamental operation, often referred to as integration. The process of determining an antiderivative can yield a family of functions differing by a constant, known as the constant of integration.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is particularly important in calculus because the derivative of ln(x) is 1/x, making it the antiderivative of 1/x. Understanding the properties of logarithms is essential for solving integrals involving exponential functions.

Integration Constant

When finding antiderivatives, the result includes an integration constant (C) because the derivative of a constant is zero. This means that multiple functions can have the same derivative, differing only by a constant value. Recognizing the importance of this constant is crucial for expressing the general solution to an indefinite integral.