To determine The improper integral without using tables. Explanation Given information: 1 θ 2 + 5 θ + 6 Formula: ∫ 1 x d x = ln | x | + C a ln | x | = ln | x a | Calculation: Express the quotient as below. 1 θ 2 + 5 θ + 6 = 1 ( θ + 2 ) ( θ + 3 ) Express the quotient as a sum of partial fractions with undetermined coefficients. 1 θ 2 + 5 θ + 6 = A ( θ + 2 ) + B ( θ + 3 ) (1) Simplify the above equation. 1 θ 2 + 5 θ + 6 = A ( θ + 3 ) + B ( θ + 2 ) ( θ + 2 ) ( θ + 3 ) 1 = A ( θ + 3 ) + B ( θ + 2 ) 1 = θ ( A + B ) + 3 A + 2 B (2) Equate the coefficient of the θ term on both the sides of equation (2). A + B = 0 (3) Equate the constant term on both the sides of equation (2). 3 A + 2 B = 1 Substitute − B for A . − 3 B + 2 B = 1 B = − 1 Substitute − 1 for B in equation (3). A = 1 Substitute − 1 for B and 1 for A in the equation (1). 1 θ 2 + 5 θ + 6 = 1 ( θ + 2 ) − 1 ( θ + 3 ) Evaluate the integral as below

Thomas' Calculus: Early Transcendentals in SI Units

14th Edition

ISBN: 9781292253114

Author: Hass, Joel R., Heil, Christopher E., WEIR, Maurice D.

Publisher: PEARSON

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Chapter 8.8, Problem 33E

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The improper integral without using tables.

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Chapter 8.8, Problem 32E

Chapter 8.8, Problem 34E

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