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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![### Calculus Question: Integral of a Function
**Problem Statement:**
What is \( f(x) = \int 3 \sqrt{x} dx \)?
**Options:**
A) \( F(x) = 2x^{\frac{3}{2}} \)
B) \( F(x) = 2x^{\frac{3}{2}} + C \)
C) \( F(x) = \frac{x^{\frac{3}{2}}}{3} + C \)
D) \( F(x) = \frac{2x^{\frac{3}{2}}}{3} + C \)
---
**Explanation of Options:**
- **Option A:** Incorrect. It represents the antiderivative without the constant of integration, which is necessary for indefinite integrals.
- **Option B:** Incorrect. While it includes the constant of integration \(C\), the coefficient with the antiderivative is not correctly simplified.
- **Option C:** Incorrect. The coefficient \( \frac{1}{3} \) is incorrect in the expression of the antiderivative.
- **Option D:** Correct. This represents the correct antiderivative of \(3 \sqrt{x}\), which simplifies to \( \frac{2x^{\frac{3}{2}}}{3} + C\).
---
#### Detailed Solution:
To find the integral \( \int 3 \sqrt{x} dx \):
1. Express \( \sqrt{x} \) as \( x^{\frac{1}{2}} \):
\[
\int 3 x^{\frac{1}{2}} dx
\]
2. Use the power rule for integration, which states \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \):
\[
\int 3 x^{\frac{1}{2}} dx = 3 \int x^{\frac{1}{2}} dx
\]
3. Apply the power rule:
\[
3 \left( \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} \right) + C = 3 \left( \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right) + C
\]
4. Simpl](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0b1d198d-95a1-4d0b-9915-fffff0f3c2fd%2Facf58728-73c0-42ae-95c1-fe90ee7dba68%2Frct3y6_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Question: Integral of a Function
**Problem Statement:**
What is \( f(x) = \int 3 \sqrt{x} dx \)?
**Options:**
A) \( F(x) = 2x^{\frac{3}{2}} \)
B) \( F(x) = 2x^{\frac{3}{2}} + C \)
C) \( F(x) = \frac{x^{\frac{3}{2}}}{3} + C \)
D) \( F(x) = \frac{2x^{\frac{3}{2}}}{3} + C \)
---
**Explanation of Options:**
- **Option A:** Incorrect. It represents the antiderivative without the constant of integration, which is necessary for indefinite integrals.
- **Option B:** Incorrect. While it includes the constant of integration \(C\), the coefficient with the antiderivative is not correctly simplified.
- **Option C:** Incorrect. The coefficient \( \frac{1}{3} \) is incorrect in the expression of the antiderivative.
- **Option D:** Correct. This represents the correct antiderivative of \(3 \sqrt{x}\), which simplifies to \( \frac{2x^{\frac{3}{2}}}{3} + C\).
---
#### Detailed Solution:
To find the integral \( \int 3 \sqrt{x} dx \):
1. Express \( \sqrt{x} \) as \( x^{\frac{1}{2}} \):
\[
\int 3 x^{\frac{1}{2}} dx
\]
2. Use the power rule for integration, which states \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \):
\[
\int 3 x^{\frac{1}{2}} dx = 3 \int x^{\frac{1}{2}} dx
\]
3. Apply the power rule:
\[
3 \left( \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} \right) + C = 3 \left( \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right) + C
\]
4. Simpl
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