Explanation Given information: x 2 ( x − 1 ) ( x 2 + 2 x + 1 ) Formula: ∫ 1 x d x = ln | x | + C a ln | x | = ln | x a | Calculation: Express the integrand as below. x 2 ( x − 1 ) ( x 2 + 2 x + 1 ) = x 2 ( x − 1 ) ( x + 1 ) 2 Express the integrand as a sum of partial fractions with undetermined coefficients. x 2 ( x − 1 ) ( x + 1 ) 2 = A x − 1 + B x + 1 + C ( x + 1 ) 2 (1) Simplify the above equation. x 2 ( x − 1 ) ( x + 1 ) 2 = [ A ( x + 1 ) 3 + B ( x − 1 ) ( x + 1 ) 2 + C ( x + 1 ) ( x − 1 ) ] ( x + 1 ) 3 ( x − 1 ) 1 = A ( x + 1 ) 2 + B ( x − 1 ) ( x + 1 ) + C ( x − 1 ) 1 = [ A ( x 2 + 1 + 2 x ) + B ( x 2 − 1 ) + C ( x − 1 ) ] 1 = x 2 ( A + B ) + x ( 2 A + C ) + A − B − C (2) Equate the coefficient of the x 2 term on both the sides of equation (2). A + B = 1 (3) Equate the coefficient of the x term on both the sides of equation (2). 2 A + C = 0 (4) Equate the constant term on both the sides of equation (2). A − B − C = 0 Substitute 1 − A for B . 2 A − C = 1 (5) Add the equations (4) and (5) as below

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ISBN: 9780134175706

Author: Unknown

Publisher: PEARSON

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Chapter 8.4, Problem 20E

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The integral through expansion of integrand as a sum of partial fractions.

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Chapter 8.4, Problem 19E

Chapter 8.4, Problem 21E

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