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...
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q8
![### Problem Statement:
Evaluate the integral:
\[ \int x^2 \cos(x^3) \, dx \]
### Solution:
To solve this integral, we can use the method of substitution.
### Step-by-Step Solution:
1. **Substitution:**
Let \( u = x^3 \). Then, \( du = 3x^2 dx \) or \( \frac{1}{3} du = x^2 dx \).
2. **Rewrite the Integral:**
Substitute \( u \) and \( du \) into the integral.
\[ \int x^2 \cos(x^3) \, dx = \int \cos(u) \cdot \frac{1}{3} \, du \]
3. **Simplify and Integrate:**
\[ \frac{1}{3} \int \cos(u) \, du \]
The integral of \( \cos(u) \) with respect to \( u \) is \( \sin(u) \). Thus,
\[ \frac{1}{3} \sin(u) + C \]
4. **Back-Substitute \( u \):**
Replace \( u \) with \( x^3 \) to get the final answer.
\[ \frac{1}{3} \sin(x^3) + C \]
### Final Answer:
\[ \int x^2 \cos(x^3) \, dx = \frac{1}{3} \sin(x^3) + C \]
Where \( C \) is the constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddf14729-dbf2-4574-b793-bebd8df9c378%2F3f9fd82c-b95c-402e-bad5-ce3172111558%2Frk3fpxo_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Evaluate the integral:
\[ \int x^2 \cos(x^3) \, dx \]
### Solution:
To solve this integral, we can use the method of substitution.
### Step-by-Step Solution:
1. **Substitution:**
Let \( u = x^3 \). Then, \( du = 3x^2 dx \) or \( \frac{1}{3} du = x^2 dx \).
2. **Rewrite the Integral:**
Substitute \( u \) and \( du \) into the integral.
\[ \int x^2 \cos(x^3) \, dx = \int \cos(u) \cdot \frac{1}{3} \, du \]
3. **Simplify and Integrate:**
\[ \frac{1}{3} \int \cos(u) \, du \]
The integral of \( \cos(u) \) with respect to \( u \) is \( \sin(u) \). Thus,
\[ \frac{1}{3} \sin(u) + C \]
4. **Back-Substitute \( u \):**
Replace \( u \) with \( x^3 \) to get the final answer.
\[ \frac{1}{3} \sin(x^3) + C \]
### Final Answer:
\[ \int x^2 \cos(x^3) \, dx = \frac{1}{3} \sin(x^3) + C \]
Where \( C \) is the constant of integration.
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