Explanation Given information: 1 x 6 ( x 5 + 4 ) Formula: ∫ 1 x d x = ln | x | + C ∫ 1 a ( x ) n d x = − 1 a ( x ) n − 1 + C Calculation: Write the integrand as below. ∫ 1 x 6 ( x 5 + 4 ) d x = ∫ x 4 x 10 ( x 5 + 4 ) d x Substitute u for x 5 and d u 5 for x 4 d x . ∫ 1 x 6 ( x 5 + 4 ) d x = ∫ 1 5 u 2 ( u + 4 ) d u Express the integrand as a sum of partial fractions with undetermined coefficients. 1 5 u 2 ( u + 4 ) = A u + B u 2 + C ( u + 4 ) (1) Simplify the above equation. 1 5 = A u ( u + 4 ) + B ( u + 4 ) + C u 2 1 5 = ( A + C ) u 2 + ( 4 A + B ) u + 4 B (2) Equate the coefficient of the u term on both the sides of equation (2). 4 A + B = 0 (3) Equate the constant term on both the sides of equation (2). 4 B = 1 5 B = 1 20 Substitute 1 20 for B in the equation (3). 4 A + 1 20 = 0 A = − 1 80 Equate the coefficient of the u 2 term on both the sides of equation (2)

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ISBN: 9781323837689

Author: WEIR

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Chapter 8.5, Problem 50E

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The integral through expressing the integrand as a sum of partial fractions.

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Chapter 8.5, Problem 49E

Chapter 8.5, Problem 51E

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