Explanation Formula used: Chain Rule: d y d x = d y d u ⋅ d u d x . Theorem: Substitution Rule for Definite Integrals. Let u = g ( x ) and g ′ ( x ) be continuous function on an interval and let f be continuous function on the corresponding range of g . On that interval ∫ a b f ( g ( x ) ) g ′ ( x ) d x = ∫ g ( a ) g ( b ) f ( u ) d u . The calculation of the ∫ 0 2 x 3 16 − x 4 d x is as follows. Notice that the integrand x 3 16 − x 4 , the function x 3 is the differentiation of the inner composite function 16 − x 4 . Consider the change of variable function as u = 16 − x 4 . Then, d u = − 4 x 3 d x . Since, the integrand is changed in terms of the variable u , the limits of integration must also be expressed in terms of u . Use the substitution u = 16 − x 4 and find the corresponding limits of u as follows. Lower limit: if x = 0 then u = 16 − 0 = 16

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 5.5, Problem 73E

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To evaluate: The value of the definite integral ∫02x316−x4 dx by using the change of variables.

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Chapter 5.5, Problem 72E

Chapter 5.5, Problem 74E

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Chapter 5 Solutions

Calculus: Early Transcendentals (2nd Edition)

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The value of the definite integral ∫ 0 2 x 3 16 − x 4 d x by using the change of variables.

Solution Summary: Theorem: Substitution Rule for Definite Integrals. The value of the definite integral is 323.

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To evaluate: The value of the definite integral ∫02x316−x4 dx by using the change of variables.

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Chapter 5 Solutions