Homework Help is Here – Start Your Trial Now!
Math
Calculus
Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.) f(x) = 6x² - 4x + 5

Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.) f(x) = 6x² - 4x + 5

Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
1st Edition
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:HOUGHTON MIFFLIN HARCOURT
Chapter8: Graphing Quadratic Functions
Section: Chapter Questions
Problem 17CT
See similar textbooks
icon
Related questions
Question
100%

5.2 Q1) Hey! I need help with the following calc question, thank you!

**Finding the Antiderivative of a Given Function** In this lesson, we'll explore how to find the most general antiderivative of a given function. We are provided with a specific function and are asked to find its antiderivative. To verify the solution, differentiation can be used. It is crucial to remember that the antiderivative includes a constant of integration, denoted here as \( C \). The given function is: \[ f(x) = 6x^2 - 4x + 5 \] **Steps to Find the Antiderivative:** 1. **Antiderivative of Each Term:** - For \( 6x^2 \): \[ \int 6x^2 \,dx = 6 \cdot \frac{x^3}{3} = 2x^3 \] - For \( -4x \): \[ \int -4x \,dx = -4 \cdot \frac{x^2}{2} = -2x^2 \] - For \( 5 \): \[ \int 5 \,dx = 5x \] 2. **Combine the Antiderivatives and Include the Constant of Integration:** \[ F(x) = 2x^3 - 2x^2 + 5x + C \] **Verification:** To verify, differentiate \( F(x) \): \[ \frac{d}{dx}(2x^3 - 2x^2 + 5x + C) = 6x^2 - 4x + 5 = f(x) \] The differentiation confirms that our antiderivative is correct. **Final Answer:** \[ F(x) = 2x^3 - 2x^2 + 5x + C \] Be sure to check your work using differentiation to ensure accuracy.
Transcribed Image Text:**Finding the Antiderivative of a Given Function** In this lesson, we'll explore how to find the most general antiderivative of a given function. We are provided with a specific function and are asked to find its antiderivative. To verify the solution, differentiation can be used. It is crucial to remember that the antiderivative includes a constant of integration, denoted here as \( C \). The given function is: \[ f(x) = 6x^2 - 4x + 5 \] **Steps to Find the Antiderivative:** 1. **Antiderivative of Each Term:** - For \( 6x^2 \): \[ \int 6x^2 \,dx = 6 \cdot \frac{x^3}{3} = 2x^3 \] - For \( -4x \): \[ \int -4x \,dx = -4 \cdot \frac{x^2}{2} = -2x^2 \] - For \( 5 \): \[ \int 5 \,dx = 5x \] 2. **Combine the Antiderivatives and Include the Constant of Integration:** \[ F(x) = 2x^3 - 2x^2 + 5x + C \] **Verification:** To verify, differentiate \( F(x) \): \[ \frac{d}{dx}(2x^3 - 2x^2 + 5x + C) = 6x^2 - 4x + 5 = f(x) \] The differentiation confirms that our antiderivative is correct. **Final Answer:** \[ F(x) = 2x^3 - 2x^2 + 5x + C \] Be sure to check your work using differentiation to ensure accuracy.
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax
Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS