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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Question
![**Problem 1: Substitution Method for Integration**
Use the Substitution Method to find the integral:
\[
\int 12x (3x^2 + 4)^9 \, dx
\]
**Choices:**
- A) \( 20(3x^2 + 4)^{10} + C \)
- B) \( 0.2(3x^2 + 4)^{10} + C \)
- C) \( 0.1(3x^2 + 4)^{10} + C \)
- D) \( 10(3x^2 + 4)^{10} + C \)
**Explanation:**
To solve this integral using substitution, let \( u = 3x^2 + 4 \). Then, differentiate \( u \) with respect to \( x \):
\[
\frac{du}{dx} = 6x \implies du = 6x \, dx
\]
Now substitute into the integral:
Replace \( 12x \, dx \) with \( 2 \, du \) (since \( 12x \, dx = 2(6x \, dx) = 2 \, du \)):
\[
\int 12x (3x^2 + 4)^9 \, dx = \int 2u^9 \, du
\]
Integrate with respect to \( u \):
\[
\int 2u^9 \, du = \frac{2}{10}u^{10} + C = 0.2u^{10} + C
\]
Finally, substitute back \( u = 3x^2 + 4 \):
\[
0.2(3x^2 + 4)^{10} + C
\]
Thus, the correct answer is choice **B**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02c8dc93-8b6f-4993-a985-0ce320a356a6%2F008e280b-f3ce-4554-8018-9ab6dba1a7c6%2Fz6vs3t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1: Substitution Method for Integration**
Use the Substitution Method to find the integral:
\[
\int 12x (3x^2 + 4)^9 \, dx
\]
**Choices:**
- A) \( 20(3x^2 + 4)^{10} + C \)
- B) \( 0.2(3x^2 + 4)^{10} + C \)
- C) \( 0.1(3x^2 + 4)^{10} + C \)
- D) \( 10(3x^2 + 4)^{10} + C \)
**Explanation:**
To solve this integral using substitution, let \( u = 3x^2 + 4 \). Then, differentiate \( u \) with respect to \( x \):
\[
\frac{du}{dx} = 6x \implies du = 6x \, dx
\]
Now substitute into the integral:
Replace \( 12x \, dx \) with \( 2 \, du \) (since \( 12x \, dx = 2(6x \, dx) = 2 \, du \)):
\[
\int 12x (3x^2 + 4)^9 \, dx = \int 2u^9 \, du
\]
Integrate with respect to \( u \):
\[
\int 2u^9 \, du = \frac{2}{10}u^{10} + C = 0.2u^{10} + C
\]
Finally, substitute back \( u = 3x^2 + 4 \):
\[
0.2(3x^2 + 4)^{10} + C
\]
Thus, the correct answer is choice **B**.
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