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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![The image contains an integral for educational purposes. The integral is presented as follows:
\[ \int \frac{x + 6}{x^7} \, dx \]
To break it down:
- The integral sign \(\int\) indicates that we are to find the antiderivative of the given function.
- The function inside the integral is \(\frac{x + 6}{x^7}\).
- The differential \(dx\) indicates that we are integrating with respect to \(x\).
This integral can be simplified by breaking it into separate terms:
\[ \int \left( \frac{x}{x^7} + \frac{6}{x^7} \right) dx = \int \left( x^{-6} + 6x^{-7} \right) dx \]
Then, we can integrate each term separately. This process involves using the power rule for integration, which states:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
where \(C\) is the constant of integration, and \(n \neq -1\).
Applying this rule:
For \( \int x^{-6} dx \) :
\[ \int x^{-6} dx = \frac{x^{-6+1}}{-6+1} + C = \frac{x^{-5}}{-5} + C = -\frac{1}{5} x^{-5} + C \]
For \( \int 6x^{-7} dx \) :
\[ \int 6x^{-7} dx = 6 \int x^{-7} dx \]
\[ 6 \int x^{-7} dx = 6 \left(\frac{x^{-7+1}}{-7+1} + C \right) = 6 \left( \frac{x^{-6}}{-6} + C \right) = -x^{-6} + C \]
Combining both results:
\[ \int \left( x^{-6} + 6x^{-7} \right) dx = -\frac{1}{5} x^{-5} - x^{-6} + C \]
This is the antiderivative or the result of the given integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44f9ebf7-05a0-44a3-89d0-aa50ed2840a0%2F93cd123c-2728-4e4b-b606-c1aa0f8559e5%2Fbvvinrno.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains an integral for educational purposes. The integral is presented as follows:
\[ \int \frac{x + 6}{x^7} \, dx \]
To break it down:
- The integral sign \(\int\) indicates that we are to find the antiderivative of the given function.
- The function inside the integral is \(\frac{x + 6}{x^7}\).
- The differential \(dx\) indicates that we are integrating with respect to \(x\).
This integral can be simplified by breaking it into separate terms:
\[ \int \left( \frac{x}{x^7} + \frac{6}{x^7} \right) dx = \int \left( x^{-6} + 6x^{-7} \right) dx \]
Then, we can integrate each term separately. This process involves using the power rule for integration, which states:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
where \(C\) is the constant of integration, and \(n \neq -1\).
Applying this rule:
For \( \int x^{-6} dx \) :
\[ \int x^{-6} dx = \frac{x^{-6+1}}{-6+1} + C = \frac{x^{-5}}{-5} + C = -\frac{1}{5} x^{-5} + C \]
For \( \int 6x^{-7} dx \) :
\[ \int 6x^{-7} dx = 6 \int x^{-7} dx \]
\[ 6 \int x^{-7} dx = 6 \left(\frac{x^{-7+1}}{-7+1} + C \right) = 6 \left( \frac{x^{-6}}{-6} + C \right) = -x^{-6} + C \]
Combining both results:
\[ \int \left( x^{-6} + 6x^{-7} \right) dx = -\frac{1}{5} x^{-5} - x^{-6} + C \]
This is the antiderivative or the result of the given integral.
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