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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![**Problem Statement:**
Find the integral of the function \((x + 7)(x - 3)\) with respect to \(x\).
**Mathematical Expression:**
\[
\int (x + 7)(x - 3) \, dx
\]
**Solution Steps:**
1. **Expand the Expression:**
First, expand the integrand:
\((x + 7)(x - 3) = x^2 - 3x + 7x - 21 = x^2 + 4x - 21\)
2. **Integrate Term by Term:**
Now, integrate each term separately:
\[
\int (x^2 + 4x - 21) \, dx = \int x^2 \, dx + \int 4x \, dx - \int 21 \, dx
\]
- **Integrate \(x^2\):**
\(\int x^2 \, dx = \frac{x^3}{3} + C_1\)
- **Integrate \(4x\):**
\(\int 4x \, dx = 2x^2 + C_2\)
- **Integrate \(-21\):**
\(\int (-21) \, dx = -21x + C_3\)
3. **Combine the Results:**
\[
\int (x^2 + 4x - 21) \, dx = \frac{x^3}{3} + 2x^2 - 21x + C
\]
Here, \(C = C_1 + C_2 + C_3\) is the constant of integration.
**Final Answer:**
\[
\int (x + 7)(x - 3) \, dx = \frac{x^3}{3} + 2x^2 - 21x + C
\]
This explanation and solution will guide students in understanding how to approach integrating polynomials by expanding and integrating term by term.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb155383f-1482-4158-bfc0-94ad91579101%2Fe29e3846-fbc9-4c3b-8c3b-3750df444e60%2F0psab8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the integral of the function \((x + 7)(x - 3)\) with respect to \(x\).
**Mathematical Expression:**
\[
\int (x + 7)(x - 3) \, dx
\]
**Solution Steps:**
1. **Expand the Expression:**
First, expand the integrand:
\((x + 7)(x - 3) = x^2 - 3x + 7x - 21 = x^2 + 4x - 21\)
2. **Integrate Term by Term:**
Now, integrate each term separately:
\[
\int (x^2 + 4x - 21) \, dx = \int x^2 \, dx + \int 4x \, dx - \int 21 \, dx
\]
- **Integrate \(x^2\):**
\(\int x^2 \, dx = \frac{x^3}{3} + C_1\)
- **Integrate \(4x\):**
\(\int 4x \, dx = 2x^2 + C_2\)
- **Integrate \(-21\):**
\(\int (-21) \, dx = -21x + C_3\)
3. **Combine the Results:**
\[
\int (x^2 + 4x - 21) \, dx = \frac{x^3}{3} + 2x^2 - 21x + C
\]
Here, \(C = C_1 + C_2 + C_3\) is the constant of integration.
**Final Answer:**
\[
\int (x + 7)(x - 3) \, dx = \frac{x^3}{3} + 2x^2 - 21x + C
\]
This explanation and solution will guide students in understanding how to approach integrating polynomials by expanding and integrating term by term.
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