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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2:47 minutes

Problem 4.9.19

Textbook Question

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

ƒ(x) = eˣ

Verified step by step guidance

Step 1: Recall the definition of an antiderivative. An antiderivative of a function ƒ(x) is a function F(x) such that F'(x) = ƒ(x).

Step 2: Identify the given function ƒ(x) = eˣ. Note that the derivative of eˣ is itself, meaning eˣ is its own antiderivative.

Step 3: Write the general form of the antiderivative. Since the derivative of a constant is zero, the antiderivative of eˣ includes an arbitrary constant C. Thus, F(x) = eˣ + C.

Step 4: Verify your result by differentiating F(x). Compute F'(x) = d/dx [eˣ + C]. Using the derivative rules, F'(x) = eˣ, which matches the original function ƒ(x).

Step 5: Conclude that all antiderivatives of ƒ(x) = eˣ are given by F(x) = eˣ + C, where C is an arbitrary constant.

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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 antiderivatives is essential for solving problems related to integration. The general form of an antiderivative includes a constant of integration, C, since the derivative of a constant is zero, leading to multiple valid antiderivatives.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. The function eˣ, where e is Euler's number (approximately 2.718), is a special case of an exponential function that has unique properties, particularly that its derivative is equal to itself. This property simplifies the process of finding antiderivatives for functions involving e.

Verification by Differentiation

Verification by differentiation involves taking the derivative of the antiderivative found to ensure it matches the original function. This step is crucial in confirming the correctness of the antiderivative. If the derivative of the antiderivative equals the original function, it validates the solution and demonstrates a proper understanding of the relationship between differentiation and integration.