To determine The integral through expressing the integrand as a sum of partial fractions. Explanation Given information: cos y sin 2 y + sin y − 6 Formula: ∫ 1 x d x = ln | x | + C Calculation: Write the integrand as below. ∫ cos y sin 2 y + sin y − 6 d y Substitute t for sin y and d t for cos y d y . ∫ d t t 2 + t − 6 Express the integrand through long division by eliminating the cubic term as below. 1 t 2 + t − 6 = 1 t 2 + 3 t − 2 t − 6 = 1 t ( t + 3 ) − 2 ( t + 3 ) = 1 ( t + 3 ) ( t − 2 ) Express the integrand as a sum of partial fractions with undetermined coefficients. 1 ( t + 3 ) ( t − 2 ) = A ( t − 2 ) + B ( t + 3 ) (1) Simplify the above equation. 1 ( t + 3 ) ( t − 2 ) = A ( t − 2 ) + B ( t + 3 ) 1 = A ( t + 3 ) + B ( t − 2 ) 1 = t ( A + B ) + 3 A − 2 B (2) Equate the constant term on both the sides of equation (2). 3 A − 2 B = 1 (3) Equate the coefficient of the t term on both the sides of equation (2). A + B = 0 (4) Multiply the equation (4) by 2 and add with the equation (3)

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ISBN: 9780134439099

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

Publisher: Pearson,

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Chapter 8.5, Problem 41E

To determine

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

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Chapter 8.5, Problem 40E

Chapter 8.5, Problem 42E

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