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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2:09 minutes

Problem 4.2.38

Textbook Question

Find all functions whose derivative is f'(x) = x + 1.

Verified step by step guidance

Start by understanding that the problem is asking for a function whose derivative is given. This means we need to find the antiderivative or integral of the function f'(x) = x + 1.

Recall that the antiderivative of a function f'(x) is a function F(x) such that F'(x) = f'(x). Therefore, we need to integrate f'(x) = x + 1 with respect to x.

To integrate f'(x) = x + 1, apply the basic rules of integration: the integral of x with respect to x is (1/2)x^2, and the integral of 1 with respect to x is x.

Combine the results from the integration: the antiderivative of x + 1 is (1/2)x^2 + x. Remember to add the constant of integration, C, because the derivative of a constant is zero, and it could be any real number.

Thus, the general solution for the function whose derivative is f'(x) = x + 1 is F(x) = (1/2)x^2 + x + 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.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It represents the slope of the tangent line to the function's graph at any given point. Understanding derivatives is crucial for solving problems related to rates of change and for finding functions from their derivatives.

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In this context, finding all functions whose derivative is f'(x) = x + 1 involves determining the antiderivative of that expression. The general form of an antiderivative includes a constant of integration, reflecting the fact that there are infinitely many functions that can share the same derivative.

Integration

Integration is the process of finding the antiderivative of a function. It is a fundamental concept in calculus that allows us to compute areas under curves and solve differential equations. In this case, integrating the function f'(x) = x + 1 will yield the original function f(x), along with a constant term that accounts for all possible vertical shifts of the function.

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