Explanation Given: The integral function is ∫ x 1 − x 4 d x Calculation: Rearrange the given function as, ∫ x 1 − x 4 d x = ∫ x 1 − ( x 2 ) 2 d x (2) Consider x 2 = sin θ (3) Differentiate both sides of the Equation (3). 2 x d x = cos θ d θ x d x = cos θ 2 d θ Substitute ( sin θ ) for ( x 2 ) and ( cos θ 2 d θ ) for ( x d x ) in Equation (2). ∫ x 1 − ( x 2 ) 2 d x = ∫ 1 − ( sin θ ) 2 ( cos θ 2 d θ ) = 1 2 ∫ 1 − sin 2 θ cos θ d θ = 1 2 ∫ cos 2 θ cos θ d θ = 1 2 ∫ cos θ cos θ d θ Simplify further as, ∫ x 1 − ( x 2 ) 2 d x = 1 2 ∫ cos 2 θ d θ = 1 2 ∫ 1 2 ( 1 + cos 2 θ ) d θ = 1 4 ∫ ( 1 + cos 2 θ ) d θ (4) Integrate Equation (4). 1 4 ∫ ( 1 + cos 2 θ ) d θ = 1 4 [ θ + 1 2 sin 2 θ ] + C = 1 4 θ + 1 8 ( 2 sin θ cos θ ) + C = 1 4 θ + 1 4 sin θ cos θ + C (5) Rearrange Equation (3)

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6.2, Problem 59E

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The integral function ∫x1−x4dx.

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Chapter 6.2, Problem 64E

Chapter 6.2, Problem 60E

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Chapter 6 Solutions

Essential Calculus: Early Transcendentals

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The integral function ∫x1−x4dx.

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Chapter 6 Solutions