Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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.
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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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