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 Statement:**
Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.)
\[
\int \frac{x + 5}{\sqrt{x}} \, dx
\]
**Solution:**
To solve this integral, we first simplify the integrand:
\[
\frac{x + 5}{\sqrt{x}} = \frac{x}{\sqrt{x}} + \frac{5}{\sqrt{x}} = x^{1/2} + 5x^{-1/2}
\]
Now, integrate term by term:
1. Integrate \(x^{1/2}\):
\[
\int x^{1/2} \, dx = \frac{x^{3/2}}{\frac{3}{2}} = \frac{2}{3}x^{3/2}
\]
2. Integrate \(5x^{-1/2}\):
\[
\int 5x^{-1/2} \, dx = 5 \cdot \frac{x^{1/2}}{\frac{1}{2}} = 10x^{1/2}
\]
Combine the results and add the constant of integration:
\[
\int \frac{x + 5}{\sqrt{x}} \, dx = \frac{2}{3}x^{3/2} + 10x^{1/2} + C
\]
**Verification by Differentiation:**
Differentiate \(\frac{2}{3}x^{3/2} + 10x^{1/2} + C\) to verify the result:
1. Differentiate \(\frac{2}{3}x^{3/2}\):
\[
\frac{d}{dx}\left(\frac{2}{3}x^{3/2}\right) = \frac{2}{3} \cdot \frac{3}{2}x^{1/2} = x^{1/2}
\]
2. Differentiate \(10x^{1/2}\):
\[
\frac{d}{dx}(10x^{1/2}) = 10 \cdot \frac{1}{2}x^{-1/2} = 5x^{-1/2}
\]
Adding these derivatives gives:
\[
x^{1/2} + 5x^{-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c34ee-e27f-46b4-a2d2-ab5f4ef78243%2F0d1faadf-5465-403e-9914-c4373c49694e%2F952k55j_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.)
\[
\int \frac{x + 5}{\sqrt{x}} \, dx
\]
**Solution:**
To solve this integral, we first simplify the integrand:
\[
\frac{x + 5}{\sqrt{x}} = \frac{x}{\sqrt{x}} + \frac{5}{\sqrt{x}} = x^{1/2} + 5x^{-1/2}
\]
Now, integrate term by term:
1. Integrate \(x^{1/2}\):
\[
\int x^{1/2} \, dx = \frac{x^{3/2}}{\frac{3}{2}} = \frac{2}{3}x^{3/2}
\]
2. Integrate \(5x^{-1/2}\):
\[
\int 5x^{-1/2} \, dx = 5 \cdot \frac{x^{1/2}}{\frac{1}{2}} = 10x^{1/2}
\]
Combine the results and add the constant of integration:
\[
\int \frac{x + 5}{\sqrt{x}} \, dx = \frac{2}{3}x^{3/2} + 10x^{1/2} + C
\]
**Verification by Differentiation:**
Differentiate \(\frac{2}{3}x^{3/2} + 10x^{1/2} + C\) to verify the result:
1. Differentiate \(\frac{2}{3}x^{3/2}\):
\[
\frac{d}{dx}\left(\frac{2}{3}x^{3/2}\right) = \frac{2}{3} \cdot \frac{3}{2}x^{1/2} = x^{1/2}
\]
2. Differentiate \(10x^{1/2}\):
\[
\frac{d}{dx}(10x^{1/2}) = 10 \cdot \frac{1}{2}x^{-1/2} = 5x^{-1/2}
\]
Adding these derivatives gives:
\[
x^{1/2} + 5x^{-
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