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Use partial fractions to find the integral. S 4-x 2x²+x-1 Step 1 Rewrite the polynomial as 2x² + x1 = 2x² - x + 2x - 1. Collect the terms according to the power of x on the right side and factorize. 2x² + 2x -x-1= x(2x - 1) + (2x - 1) = (2x - 1)(x + 1) Because 2x² + x - 1 = (2x − 1)(x + 1), you should include one partial fraction for each factor and write B Step 2 4-x 2x²+x-1 dx where A and B are to be determined. 2x To solve for A, let x = 1/2✔ 4 2 Multiplying this equation by the least common denominator (2x - 1)(x + 1) yields the basic equation 4-x= A(x+1 + B(2x - 1). x+1 Because this equation is to be true for all x, substitute any convenient values for x to obtain equations in A and B. Step 3 The most convenient values are the ones that make particular factors equal to 0. Step 4 To solve for B, let x = 4 1/2 and obtain 4 4 − ½⁄2 = A( 1⁄2 + ¹) + B( 2 × ½1⁄2-1) 1x A = 7/3 ✔ 4 + 1 = A(-1 X =4 2x B = -5/3 2 1 7/3 X X X X to obtain the following. + 1) + B(2(-1 0 + B(4 1 x ) ✓ ) - 1)

Use partial fractions to find the integral. S 4-x 2x²+x-1 Step 1 Rewrite the polynomial as 2x² + x1 = 2x² - x + 2x - 1. Collect the terms according to the power of x on the right side and factorize. 2x² + 2x -x-1= x(2x - 1) + (2x - 1) = (2x - 1)(x + 1) Because 2x² + x - 1 = (2x − 1)(x + 1), you should include one partial fraction for each factor and write B Step 2 4-x 2x²+x-1 dx where A and B are to be determined. 2x To solve for A, let x = 1/2✔ 4 2 Multiplying this equation by the least common denominator (2x - 1)(x + 1) yields the basic equation 4-x= A(x+1 + B(2x - 1). x+1 Because this equation is to be true for all x, substitute any convenient values for x to obtain equations in A and B. Step 3 The most convenient values are the ones that make particular factors equal to 0. Step 4 To solve for B, let x = 4 1/2 and obtain 4 4 − ½⁄2 = A( 1⁄2 + ¹) + B( 2 × ½1⁄2-1) 1x A = 7/3 ✔ 4 + 1 = A(-1 X =4 2x B = -5/3 2 1 7/3 X X X X to obtain the following. + 1) + B(2(-1 0 + B(4 1 x ) ✓ ) - 1)

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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Use partial fractions to find the integral. 4- X | 2x² 2x² + x1 Step 1 Rewrite the polynomial as 2x² + x1 = 2x² - x + 2x - 1. Collect the terms according to the power of x on the right side and factorize. 2x² + 2x -x-1 = x(2x − 1) + (2x - 1) = (2x - 1)(x + 1) Because 2x² + x − 1 = (2x − 1)(x + 1), you should include one partial fraction for each factor and write A 4-X 2x²+x-1 where A and B are to be determined. dx To solve for A, let x = 1/2✔ 1 x 2 4 2x Step 2 Multiplying this equation by the least common denominator (2x - 1)(x + 1) yields the basic equation 4x = A(x+1 + B(2x - 1). Step 3 The most convenient values are the ones that make particular factors equal to 0. 7 x+1 Because this equation is to be true for all x, substitute any convenient values for x to obtain equations in A and B. Step 4 To solve for B, let x = 4 1/2 and obtain 4 − —⁄2 = A ( ²¹1 + 1) + B( ² × 1⁄2 − 1) 2 - 2 A = 7/3 X = AX 4 + 1 = A(-1 2x = 4 B = -5/3 1 2 + 5 x 7/3 x X to obtain the following. + 1) + B(2(-1 B X + B(4 X × ) + 1 ✔)-1)
Transcribed Image Text:Use partial fractions to find the integral. 4- X | 2x² 2x² + x1 Step 1 Rewrite the polynomial as 2x² + x1 = 2x² - x + 2x - 1. Collect the terms according to the power of x on the right side and factorize. 2x² + 2x -x-1 = x(2x − 1) + (2x - 1) = (2x - 1)(x + 1) Because 2x² + x − 1 = (2x − 1)(x + 1), you should include one partial fraction for each factor and write A 4-X 2x²+x-1 where A and B are to be determined. dx To solve for A, let x = 1/2✔ 1 x 2 4 2x Step 2 Multiplying this equation by the least common denominator (2x - 1)(x + 1) yields the basic equation 4x = A(x+1 + B(2x - 1). Step 3 The most convenient values are the ones that make particular factors equal to 0. 7 x+1 Because this equation is to be true for all x, substitute any convenient values for x to obtain equations in A and B. Step 4 To solve for B, let x = 4 1/2 and obtain 4 − —⁄2 = A ( ²¹1 + 1) + B( ² × 1⁄2 − 1) 2 - 2 A = 7/3 X = AX 4 + 1 = A(-1 2x = 4 B = -5/3 1 2 + 5 x 7/3 x X to obtain the following. + 1) + B(2(-1 B X + B(4 X × ) + 1 ✔)-1)
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