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True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $f^{\prime}(x)=g(x),$ then $\int g(x) d x=f(x)+C$.

Name: Problem 93, Chapter 4: Calculus of a Single Variable
Uploaded: 2020-10-22T02:01:30-08:00
Duration: 1 min 7 s
Channel: Gregory Higby
Description: True or False? In Exercises 90-95 , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f^'(x)=g(x), then ∫g(x) d x=f(x)+C.

   True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $f^{\prime}(x)=g(x),$ then $\int g(x) d x=f(x)+C$.

Calculus of a Single Variable

Ron Larson, Bruce… 9th Edition

Chapter 4, Problem 93

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Solved by Expert Gregory Higby

10/22/2020

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This is essentially the definition of an antiderivative. Show more…

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True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $f^{\prime}(x)=g(x),$ then $\int g(x) d x=f(x)+C$.

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Key Concepts

Fundamental Theorem of Calculus

This theorem establishes that integration and differentiation are inverse processes. In particular, if a function f is differentiable and its derivative is given by g, then the integral of g retrieves the original function f up to an arbitrary constant, which connects the processes of accumulation and rate of change.

Antiderivatives

An antiderivative of a function g(x) is any function f(x) such that f'(x) = g(x). The statement in the question relies on this concept, asserting that integrating g(x) gives f(x) plus a constant, where f is any antiderivative of g.

Constant of Integration

When taking the indefinite integral of a function, an arbitrary constant C is added because differentiation of any constant yields 0. This accounts for all functions that differ only by a constant, hence the notation f(x) + C in the result.

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