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
![### Find the Antiderivative
Evaluate the integral:
\[
\int \frac{10x^3 - 5x^2 + 7}{x^6} \, dx
\]
**Solution Process:**
1. **Simplify the Expression:**
- Divide each term in the numerator by \(x^6\):
\[
\int \left( \frac{10x^3}{x^6} - \frac{5x^2}{x^6} + \frac{7}{x^6} \right) \, dx = \int \left( 10x^{-3} - 5x^{-4} + 7x^{-6} \right) \, dx
\]
2. **Find the Antiderivative of Each Term:**
- Use the power rule for integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
- Antiderivative of \(10x^{-3}\):
\[
10 \cdot \frac{x^{-2}}{-2} = -5x^{-2}
\]
- Antiderivative of \(-5x^{-4}\):
\[
-5 \cdot \frac{x^{-3}}{-3} = \frac{5}{3}x^{-3}
\]
- Antiderivative of \(7x^{-6}\):
\[
7 \cdot \frac{x^{-5}}{-5} = -\frac{7}{5}x^{-5}
\]
3. **Combine the Results:**
\[
-5x^{-2} + \frac{5}{3}x^{-3} - \frac{7}{5}x^{-5} + C
\]
This results in the complete antiderivative:
\[
\int \frac{10x^3 - 5x^2 + 7}{x^6} \, dx = -5x^{-2} + \frac{5}{3}x^{-3} - \frac{7}{5}x^{-5} + C
\]
where \(C\) is the constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b638ef2-9928-44ee-907b-a6bc7b9add08%2F03635eae-f2b4-4dc0-bff9-5b8af5557dfb%2Ffrg71o4_processed.png&w=3840&q=75)
Transcribed Image Text:### Find the Antiderivative
Evaluate the integral:
\[
\int \frac{10x^3 - 5x^2 + 7}{x^6} \, dx
\]
**Solution Process:**
1. **Simplify the Expression:**
- Divide each term in the numerator by \(x^6\):
\[
\int \left( \frac{10x^3}{x^6} - \frac{5x^2}{x^6} + \frac{7}{x^6} \right) \, dx = \int \left( 10x^{-3} - 5x^{-4} + 7x^{-6} \right) \, dx
\]
2. **Find the Antiderivative of Each Term:**
- Use the power rule for integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
- Antiderivative of \(10x^{-3}\):
\[
10 \cdot \frac{x^{-2}}{-2} = -5x^{-2}
\]
- Antiderivative of \(-5x^{-4}\):
\[
-5 \cdot \frac{x^{-3}}{-3} = \frac{5}{3}x^{-3}
\]
- Antiderivative of \(7x^{-6}\):
\[
7 \cdot \frac{x^{-5}}{-5} = -\frac{7}{5}x^{-5}
\]
3. **Combine the Results:**
\[
-5x^{-2} + \frac{5}{3}x^{-3} - \frac{7}{5}x^{-5} + C
\]
This results in the complete antiderivative:
\[
\int \frac{10x^3 - 5x^2 + 7}{x^6} \, dx = -5x^{-2} + \frac{5}{3}x^{-3} - \frac{7}{5}x^{-5} + C
\]
where \(C\) is the constant of integration.
![**Problem Statement:**
Find the integral of the expression:
\[
\int \left( -3x^6 + \frac{6}{x} - \frac{3}{x^6} + \sqrt{x} \right) \, dx
\]
After evaluating the integral, include a constant of integration \( C \).
---
**Explanation:**
The given expression is a polynomial combined with rational and radical terms. To integrate, each term should be approached using basic integral formulas:
1. **Polynomial Integration:** Integrate \( x^n \) as \( \frac{x^{n+1}}{n+1} \).
2. **Rational Integration:** For terms like \( \frac{1}{x^n} \), rewrite as \( x^{-n} \) and then apply the polynomial rule.
3. **Radical Integration:** Rewrite \( \sqrt{x} \) as \( x^{1/2} \) and integrate.
The solution will include a constant of integration, denoted as \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b638ef2-9928-44ee-907b-a6bc7b9add08%2F03635eae-f2b4-4dc0-bff9-5b8af5557dfb%2F8kbifug_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the integral of the expression:
\[
\int \left( -3x^6 + \frac{6}{x} - \frac{3}{x^6} + \sqrt{x} \right) \, dx
\]
After evaluating the integral, include a constant of integration \( C \).
---
**Explanation:**
The given expression is a polynomial combined with rational and radical terms. To integrate, each term should be approached using basic integral formulas:
1. **Polynomial Integration:** Integrate \( x^n \) as \( \frac{x^{n+1}}{n+1} \).
2. **Rational Integration:** For terms like \( \frac{1}{x^n} \), rewrite as \( x^{-n} \) and then apply the polynomial rule.
3. **Radical Integration:** Rewrite \( \sqrt{x} \) as \( x^{1/2} \) and integrate.
The solution will include a constant of integration, denoted as \( C \).
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