Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$\int \frac{d}{d x}[f(x)] d x=f(x)$$

Name: Problem 99, Chapter 6: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-05-10T15:12:01-08:00
Duration: 1 min 36 s
Channel: Gregory Higby
Description: Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ∫(d)/(d x)[f(x)] d x=f(x)

   Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$\int \frac{d}{d x}[f(x)] d x=f(x)$$

Applied Calculus for the MLSS A Brief Approach

Soo T. Tan 10th Edition

Chapter 6, Problem 99

Instant Answer

Solved by Expert Gregory Higby

05/10/2021

Step 1

The statement is asking whether the integral of the derivative of a function $ f(x) $ with respect to $ x $ is equal to the original function $ f(x) $. Show more…

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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$\int \frac{d}{d x}[f(x)] d x=f(x)$$

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Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration by stating that integration reverses the process of differentiation. More specifically, if you differentiate a function f(x) and then integrate that derivative, you recover the original function f(x) up to an additive constant. This theorem is central in understanding how integration and differentiation are inverse operations.

Indefinite Integral and Constant of Integration

An indefinite integral represents a family of functions that differ by a constant term because the derivative of any constant is zero. Therefore, when you take the indefinite integral of the derivative of f(x), the result is f(x) + C, where C is an arbitrary constant. Neglecting this constant can lead to misleading conclusions about the equality between the integrated function and the original function.

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