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. If $f$ and $g$ are integrable, then $\int[2 f(x)-3 g(x)] d x=$ $2 \int f(x) d x-3 \int g(x) d x$

Name: Problem 99, Chapter 11: Applied Mathematics: For the Managerial, Life, and Social Sciences
Uploaded: 2021-05-10T15:17:43-08:00
Duration: 1 min 45 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. If f and g are integrable, then ∫[2 f(x)-3 g(x)] d x= 2 ∫f(x) d x-3 ∫g(x) d 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.
If $f$ and $g$ are integrable, then $\int[2 f(x)-3 g(x)] d x=$ $2 \int f(x) d x-3 \int g(x) d x$

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan 5th Edition

Chapter 11, Problem 99

Instant Answer

Solved by Expert Gregory Higby

05/10/2021

Step 1

The statement is true. To explain why it is true, we can use the linearity property of integrals. The linearity property states that for any constants $a$ and $b$, and any integrable functions $f(x)$ and $g(x)$, we have: $$\int [a f(x) + b g(x)] dx = a \int Show more…

Show all steps

Thanks for your feedback!

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. If $f$ and $g$ are integrable, then $\int[2 f(x)-3 g(x)] d x=$ $2 \int f(x) d x-3 \int g(x) d x$

Play audio

Feedback

Gregory Higby and 101 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Linearity of Integration

Integration is a linear operation, which means it obeys both the additivity property and the constant multiple property. This principle ensures that when you take the integral of a sum or difference of functions, multiplied by constants, it is equivalent to taking the corresponding linear combination of the integrals of those functions. This is exactly the property described in the statement.

Integrability

For the linearity of integration to hold, the functions involved must be integrable; that is, their integrals must exist over the specified domain. In this case, the assumption that f and g are integrable guarantees that their integrals are well-defined, thereby allowing the application of the linearity properties of the integral.

Please give Ace some feedback