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
100%
![### Integration Substitution Example
**Problem Statement:**
Evaluate the integral
\[ \int x^3 (x^4 - 11)^{27} \, dx \]
by making the substitution \( u = x^4 - 11 \).
**Detailed Solution:**
To solve the integral
\[ \int x^3 (x^4 - 11)^{27} \, dx \]
we will use the substitution method. Let's use the substitution \( u = x^4 - 11 \).
1. **Substitute the expression:**
First, identify \( u \):
\[ u = x^4 - 11 \]
2. **Differentiate \( u \) with respect to \( x \):**
\[ \frac{du}{dx} = 4x^3 \]
3. **Solve for \( dx \):**
\[ du = 4x^3 \, dx \]
\[ dx = \frac{du}{4x^3} \]
4. **Substitute \( u \) and \( dx \) in the integral:**
Rewrite the integral in terms of \( u \):
\[ \int x^3 (x^4 - 11)^{27} \, dx \]
Based on \( u = x^4 - 11 \):
\[ \int x^3 u^{27} \frac{du}{4x^3} \]
5. **Simplify the integral:**
\[ \int \frac{1}{4} u^{27} \, du \]
\[ \frac{1}{4} \int u^{27} \, du \]
6. **Integrate with respect to \( u \):**
\[ \frac{1}{4} \cdot \frac{u^{28}}{28} = \frac{u^{28}}{112} \]
7. **Substitute back \( u = x^4 - 11 \):**
\[ \frac{(x^4 - 11)^{28}}{112} + C \]
**Final Answer:**
\[ \frac{(x^4 - 11)^{28}}{112} + C \]
**Note**: Your answer should be in terms of \( x \) and not \( u \).
This method of substitution](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb53d735-0a0d-4695-97c8-e09c334fb263%2F14794eb6-67cd-4687-b293-39e6fe81a423%2F0fvghss_processed.png&w=3840&q=75)
Transcribed Image Text:### Integration Substitution Example
**Problem Statement:**
Evaluate the integral
\[ \int x^3 (x^4 - 11)^{27} \, dx \]
by making the substitution \( u = x^4 - 11 \).
**Detailed Solution:**
To solve the integral
\[ \int x^3 (x^4 - 11)^{27} \, dx \]
we will use the substitution method. Let's use the substitution \( u = x^4 - 11 \).
1. **Substitute the expression:**
First, identify \( u \):
\[ u = x^4 - 11 \]
2. **Differentiate \( u \) with respect to \( x \):**
\[ \frac{du}{dx} = 4x^3 \]
3. **Solve for \( dx \):**
\[ du = 4x^3 \, dx \]
\[ dx = \frac{du}{4x^3} \]
4. **Substitute \( u \) and \( dx \) in the integral:**
Rewrite the integral in terms of \( u \):
\[ \int x^3 (x^4 - 11)^{27} \, dx \]
Based on \( u = x^4 - 11 \):
\[ \int x^3 u^{27} \frac{du}{4x^3} \]
5. **Simplify the integral:**
\[ \int \frac{1}{4} u^{27} \, du \]
\[ \frac{1}{4} \int u^{27} \, du \]
6. **Integrate with respect to \( u \):**
\[ \frac{1}{4} \cdot \frac{u^{28}}{28} = \frac{u^{28}}{112} \]
7. **Substitute back \( u = x^4 - 11 \):**
\[ \frac{(x^4 - 11)^{28}}{112} + C \]
**Final Answer:**
\[ \frac{(x^4 - 11)^{28}}{112} + C \]
**Note**: Your answer should be in terms of \( x \) and not \( u \).
This method of substitution
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781319050740/9781319050740_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
![Precalculus](https://www.bartleby.com/isbn_cover_images/9780135189405/9780135189405_smallCoverImage.gif)
![Calculus: Early Transcendental Functions](https://www.bartleby.com/isbn_cover_images/9781337552516/9781337552516_smallCoverImage.gif)
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning