Write the integral as the sum of the integral of an odd function and the integral of an even function. [₁₁²+72- +7x²5x8) dx 10 [₁(x²³ - 8) dx + [ { (7x² - 5x) dx O • L₁, (x² - 5x) dx + L₁ (7x² - 8 dx 3 01₁ (x² - 5x - 8) dx + 1₁7x² 7x² dx • 1³₁ x³ dx + 1²₁ (7x² 0 L₁(x² + 7x²) dx + L₁ (- (7x²5x - 8) dx (-5x - 8) dx - 8) dx Use this simplification to evaluate the integral. -90
Write the integral as the sum of the integral of an odd function and the integral of an even function. [₁₁²+72- +7x²5x8) dx 10 [₁(x²³ - 8) dx + [ { (7x² - 5x) dx O • L₁, (x² - 5x) dx + L₁ (7x² - 8 dx 3 01₁ (x² - 5x - 8) dx + 1₁7x² 7x² dx • 1³₁ x³ dx + 1²₁ (7x² 0 L₁(x² + 7x²) dx + L₁ (- (7x²5x - 8) dx (-5x - 8) dx - 8) dx Use this simplification to evaluate the integral. -90
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Title: Decomposing Integrals into Odd and Even Functions**
**Objective:**
Learn to write the integral of a polynomial function as the sum of the integrals of an odd function and an even function.
**Problem:**
Evaluate the following integral:
\[
\int_{-3}^{3} (x^3 + 7x^2 - 5x - 8) \, dx
\]
**Approach:**
We aim to express this integral as the sum of two separate integrals: one for an odd function and one for an even function.
**Options:**
1. \(\int_{-3}^{3} (x^3 - 8) \, dx + \int_{-3}^{3} (7x^2 - 5x) \, dx\)
2. \(\int_{-3}^{3} (x^3 - 5x) \, dx + \int_{-3}^{3} (7x^2 - 8) \, dx\) *(Selected Option)*
3. \(\int_{-3}^{3} (x^3 - 5x - 8) \, dx + \int_{-3}^{3} 7x^2 \, dx\)
4. \(\int_{-3}^{3} x^3 \, dx + \int_{-3}^{3} (7x^2 - 5x - 8) \, dx\)
5. \(\int_{-3}^{3} (x^3 + 7x^2) \, dx + \int_{-3}^{3} (-5x - 8) \, dx\) *(Correct Answer)*
**Solution:**
Using the selected correct option:
\[
\int_{-3}^{3} (x^3 + 7x^2) \, dx + \int_{-3}^{3} (-5x - 8) \, dx
\]
**Evaluation:**
By symmetry properties of odd and even functions, simplify the integral:
\[
\boxed{-90}
\]
This approach breaks the integral into components, utilizing the properties of odd and even functions for efficient evaluation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf9dafc6-bac8-41ec-a856-62d58445bc24%2F977493f2-f87c-47ba-9a71-ced26ca607d1%2Fd4gthqh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Decomposing Integrals into Odd and Even Functions**
**Objective:**
Learn to write the integral of a polynomial function as the sum of the integrals of an odd function and an even function.
**Problem:**
Evaluate the following integral:
\[
\int_{-3}^{3} (x^3 + 7x^2 - 5x - 8) \, dx
\]
**Approach:**
We aim to express this integral as the sum of two separate integrals: one for an odd function and one for an even function.
**Options:**
1. \(\int_{-3}^{3} (x^3 - 8) \, dx + \int_{-3}^{3} (7x^2 - 5x) \, dx\)
2. \(\int_{-3}^{3} (x^3 - 5x) \, dx + \int_{-3}^{3} (7x^2 - 8) \, dx\) *(Selected Option)*
3. \(\int_{-3}^{3} (x^3 - 5x - 8) \, dx + \int_{-3}^{3} 7x^2 \, dx\)
4. \(\int_{-3}^{3} x^3 \, dx + \int_{-3}^{3} (7x^2 - 5x - 8) \, dx\)
5. \(\int_{-3}^{3} (x^3 + 7x^2) \, dx + \int_{-3}^{3} (-5x - 8) \, dx\) *(Correct Answer)*
**Solution:**
Using the selected correct option:
\[
\int_{-3}^{3} (x^3 + 7x^2) \, dx + \int_{-3}^{3} (-5x - 8) \, dx
\]
**Evaluation:**
By symmetry properties of odd and even functions, simplify the integral:
\[
\boxed{-90}
\]
This approach breaks the integral into components, utilizing the properties of odd and even functions for efficient evaluation.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning