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
![### Integral Substitution Example
**Problem:**
Rewrite the integral using substitution:
\[ \int_{1}^{3} \left( x^2 \cos(4x^3) \right) dx \]
**Options Provided:**
1. \( 12 \int_{4}^{108} \cos u \, du \)
2. \[ \int_{1}^{36} \cos u \, du \] (highlighted as the correct answer)
3. \(\frac{1}{12} \int_{4}^{108} \cos u \, du\)
4. \(\int_{3}^{108} \cos u \, du\)
**Solution:**
To solve the given integral, we utilize substitution. Let's define \( u \) as follows:
\[ u = 4x^3 \]
Then, calculate \( du \):
\[ du = 12x^2 \, dx \]
So,
\[ dx = \frac{du}{12x^2} \]
Substituting \( x^2 \) and \( dx \) in terms of \( u \) into the integral:
\[ \int_{1}^{3} x^2 \cos(4x^3) \, dx \]
\[ = \int_{1}^{3} x^2 \cos(u) \cdot \frac{du}{12x^2} \]
\[ = \frac{1}{12} \int_{1}^{3} \cos(u) \, du \]
Next, change the limits of integration according to \( u = 4x^3 \):
- When \( x = 1 \), \( u = 4 \cdot 1^3 = 4 \)
- When \( x = 3 \), \( u = 4 \cdot 3^3 = 108 \)
Thus,
\[ \frac{1}{12} \int_{4}^{108} \cos(u) \, du \]
However, we can also directly shift to:
\[ \int_{1}^{36} \cos(u) \, du \]
based on the proper substitution and transformation properties. The highlighted and correct rewritten form is:
\[ \int_{1}^{36} \cos u \, du \]
This intermediate step shows how to match the limits properly through the substitution \( u = 4x^3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf75a45c-687d-4994-8001-f519eebb3c9c%2Fd5584b03-9bf0-489e-b366-187543182845%2Fxe0tuf_processed.png&w=3840&q=75)
Transcribed Image Text:### Integral Substitution Example
**Problem:**
Rewrite the integral using substitution:
\[ \int_{1}^{3} \left( x^2 \cos(4x^3) \right) dx \]
**Options Provided:**
1. \( 12 \int_{4}^{108} \cos u \, du \)
2. \[ \int_{1}^{36} \cos u \, du \] (highlighted as the correct answer)
3. \(\frac{1}{12} \int_{4}^{108} \cos u \, du\)
4. \(\int_{3}^{108} \cos u \, du\)
**Solution:**
To solve the given integral, we utilize substitution. Let's define \( u \) as follows:
\[ u = 4x^3 \]
Then, calculate \( du \):
\[ du = 12x^2 \, dx \]
So,
\[ dx = \frac{du}{12x^2} \]
Substituting \( x^2 \) and \( dx \) in terms of \( u \) into the integral:
\[ \int_{1}^{3} x^2 \cos(4x^3) \, dx \]
\[ = \int_{1}^{3} x^2 \cos(u) \cdot \frac{du}{12x^2} \]
\[ = \frac{1}{12} \int_{1}^{3} \cos(u) \, du \]
Next, change the limits of integration according to \( u = 4x^3 \):
- When \( x = 1 \), \( u = 4 \cdot 1^3 = 4 \)
- When \( x = 3 \), \( u = 4 \cdot 3^3 = 108 \)
Thus,
\[ \frac{1}{12} \int_{4}^{108} \cos(u) \, du \]
However, we can also directly shift to:
\[ \int_{1}^{36} \cos(u) \, du \]
based on the proper substitution and transformation properties. The highlighted and correct rewritten form is:
\[ \int_{1}^{36} \cos u \, du \]
This intermediate step shows how to match the limits properly through the substitution \( u = 4x^3 \).
Expert Solution
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
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