Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 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 Integrals3h 25m
7. Antiderivatives & Indefinite Integrals
Antiderivatives
Problem 4.2.38
Textbook Question
Find all functions whose derivative is f'(x) = x + 1.
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1
Recognize that to find the functions whose derivative is given, we need to perform the process of integration on the derivative f'(x) = x + 1.
Set up the integral: f(x) = ∫(x + 1) dx, which means we will integrate the expression x + 1 with respect to x.
Apply the power rule of integration to each term: for the term x, the integral is (1/2)x^2, and for the constant term 1, the integral is x.
Combine the results of the integration: f(x) = (1/2)x^2 + x + C, where C is the constant of integration that represents any constant value that could be added.
Conclude that the general solution for the function f(x) is f(x) = (1/2)x^2 + x + C, where C can be any real number.
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