Explanation Given information: ∫ d x ( 9 − x 2 ) Formula: ∫ 1 x d x = ln | x | + C Calculation: Evaluate the integral as given below. ∫ d x ( 9 − x 2 ) = ∫ d x ( 3 2 − x 2 ) ∫ d x ( 9 − x 2 ) = ∫ 1 ( 3 − x ) ( 3 + x ) d x (1) Express the integrand as a sum of partial fractions with undetermined coefficients. 1 ( 3 − x ) ( 3 + x ) = A 3 − x + B 3 + x (2) Simplify the above equation. 1 ( 3 − x ) ( 3 + x ) = A ( 3 + x ) + B ( 3 − x ) ( 3 − x ) ( 3 + x ) 1 = A ( 3 + x ) + B ( 3 − x ) 1 = 3 A + 3 B + x ( A − B ) (3) Equate the coefficient of the x term on both the sides of equation (3). A − B = 0 (4) Equate the constant term on both the sides of equation (3). 3 A + 3 B = 1 (5) Multiply the equation (4) by 3 and add it with the equation (5). 6 A = 1 A = 1 6 Substitute 1 6 for A in equation (4)

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 8, Problem 35PE

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