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...
Related questions
Question
Find the antiderivative
![The image presents a mathematical expression representing an integral. The expression is:
\[
\int 4x(5x^3 - 3x) \, dx
\]
This integral requires expanding the polynomial within the parentheses and integrating term by term. Here’s a breakdown of the expression:
1. **Polynomial Expansion:**
- Distribute \(4x\) across the terms inside the parentheses:
\[
4x \cdot 5x^3 = 20x^4
\]
\[
4x \cdot (-3x) = -12x^2
\]
2. **Rewritten Integral:**
\[
\int (20x^4 - 12x^2) \, dx
\]
3. **Integration:**
- For each term, increase the power by one and divide by the new power:
\[
\int 20x^4 \, dx = \frac{20}{5}x^5 = 4x^5
\]
\[
\int -12x^2 \, dx = \frac{-12}{3}x^3 = -4x^3
\]
4. **Final Result:**
\[
4x^5 - 4x^3 + C
\]
Where \(C\) is the constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fafdf4ca0-9c99-4b1f-b620-5fa5bf870b38%2F75e09690-3829-49b6-adb5-a2c42728edb9%2Fx452ggq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image presents a mathematical expression representing an integral. The expression is:
\[
\int 4x(5x^3 - 3x) \, dx
\]
This integral requires expanding the polynomial within the parentheses and integrating term by term. Here’s a breakdown of the expression:
1. **Polynomial Expansion:**
- Distribute \(4x\) across the terms inside the parentheses:
\[
4x \cdot 5x^3 = 20x^4
\]
\[
4x \cdot (-3x) = -12x^2
\]
2. **Rewritten Integral:**
\[
\int (20x^4 - 12x^2) \, dx
\]
3. **Integration:**
- For each term, increase the power by one and divide by the new power:
\[
\int 20x^4 \, dx = \frac{20}{5}x^5 = 4x^5
\]
\[
\int -12x^2 \, dx = \frac{-12}{3}x^3 = -4x^3
\]
4. **Final Result:**
\[
4x^5 - 4x^3 + C
\]
Where \(C\) is the constant of integration.
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