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 . Explanation: The calculation of the ∫ 0 1 x 1 − x 2 d x is as follows, Notice that the integrand x 1 − x 2 , the function x is the differentiation of the inner composite function 1 − x 2 . Consider the change of variable function as u = 1 − x 2 . Then, d u = − 2 x d x

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Chapter 5.5, Problem 69E

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To evaluate: The value of the definite integral ∫01x1−x2 dx by using the change of variables.

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Chapter 5.5, Problem 68E

Chapter 5.5, Problem 70E

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The value of the definite integral ∫ 0 1 x 1 − x 2 d x by using the change of variables.

Solution Summary: Theorem: Substitution Rule for Definite Integrals. The value of the definite integral is 13.

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To evaluate: The value of the definite integral ∫01x1−x2 dx by using the change of variables.

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Chapter 5 Solutions