Explanation Given information: ∫ 0 2 x e − ( 1 / 2 ) x 2 d x Formula Used: Change of limits rule: Suppose that the integral f ( g ( x ) ) g ′ ( x ) d x is subjected to the substitution u = g ( x ) so that f ( g ( x ) ) g ′ ( x ) d x becomes ∫ f ( u ) d u . Then, ∫ a b f ( g ( x ) ) g ' ( x ) d x = ∫ g ( a ) g ( b ) f ( u ) d u . Calculation: Let g ( x ) = u = − 1 2 x 2 . Differentiate with respect to x . d u d x = − 1 2 ( 2 x ) = − x ⇒ − d u = x d x Apply the substitution to the limits of integration. When x = 0 , then u = − 1 2 ( 0 ) 2 = 0 ; and when x = 2 , then u = − 1 2 ( 2 ) 2 = − 2

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Author: Larry Goldstein, David Lay

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Chapter 9, Problem 39RE

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To calculate: ∫02x e−(1/2)x2dx.

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Chapter 9, Problem 38RE

Chapter 9, Problem 40RE

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∫ 0 2 x e − ( 1 / 2 ) x 2 d x .

Solution Summary: The author explains how to change the limits of integration.

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To calculate: ∫02x e−(1/2)x2dx.

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Chapter 9 Solutions