Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### 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.
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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 \( C \).
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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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