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Use the Method of Partial Fractions to set up a new integral equal to the indefinite integral below. Do not solve the integral. 5 dx 3x3 + 12x

Use the Method of Partial Fractions to set up a new integral equal to the indefinite integral below. Do not solve the integral. 5 dx 3x3 + 12x

Calculus: Early Transcendentals
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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**Instructions:** Use the Method of Partial Fractions to set up a new integral equal to the indefinite integral below. **Do not solve the integral.** \[ \int \frac{5}{3x^3 + 12x} \, dx \] **Explanation:** The task involves rewriting the given integral in a form that facilitates its evaluation using the Method of Partial Fractions. The denominator, \(3x^3 + 12x\), should be factored first to identify the partial fractions. **Steps for Partial Fractions:** 1. **Factor the Denominator:** - Factor out the greatest common factor: \(3x^3 + 12x = 3x(x^2 + 4)\). 2. **Form of Partial Fractions:** - The expression can be written as \( \frac{A}{x} + \frac{Bx + C}{x^2 + 4} \). 3. **Set Up the Equations:** - Using the form above, equate it to the original fraction and solve for the constants \(A\), \(B\), and \(C\). These steps are essential to set up the new integral(s) but remember, **do not proceed to solve the integral** as per the instructions provided.
Transcribed Image Text:**Instructions:** Use the Method of Partial Fractions to set up a new integral equal to the indefinite integral below. **Do not solve the integral.** \[ \int \frac{5}{3x^3 + 12x} \, dx \] **Explanation:** The task involves rewriting the given integral in a form that facilitates its evaluation using the Method of Partial Fractions. The denominator, \(3x^3 + 12x\), should be factored first to identify the partial fractions. **Steps for Partial Fractions:** 1. **Factor the Denominator:** - Factor out the greatest common factor: \(3x^3 + 12x = 3x(x^2 + 4)\). 2. **Form of Partial Fractions:** - The expression can be written as \( \frac{A}{x} + \frac{Bx + C}{x^2 + 4} \). 3. **Set Up the Equations:** - Using the form above, equate it to the original fraction and solve for the constants \(A\), \(B\), and \(C\). These steps are essential to set up the new integral(s) but remember, **do not proceed to solve the integral** as per the instructions provided.
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