Explanation Given information: The improper integral is ∫ 0 ∞ x 2 e − x 3 d x . Explanation: Definition: If a be a fixed number such that f ( x ) is a nonnegative function for x ≥ a and if lim b → ∞ ∫ a b f ( x ) d x = L , then ∫ a ∞ f ( x ) d x = L . Take b > 0 and solve the integration by using the substitution method. Let u = − x 3 . Differentiate with respect to x . ⇒ d u d x = − 3 x 2 ⇒ − d u 3 = x 2 d x Apply the substitution to the limits of integration. When x = 0 , then u = − ( 0 ) 3 = 0 ; and when x = b , then u = − b 3 . Therefore, the integral becomes, ∫ 0 ∞ x 2 e − x 3 d x = ∫ 0 − b 3 e u ( − d u 3 ) = − 1 3 ∫ 0 − b 3 e u d u = − 1 3 ( e u | 0 − b 3 ) = − 1 3 ( e − b 3 − e 0 )

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Chapter 9, Problem 50RE

To determine

Whether the improper integral ∫0∞x2e−x3 dx is convergent or not, and find its

value if it is convergent.

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Chapter 9, Problem 49RE

Chapter 9, Problem 51RE

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Whether the improper integral ∫ 0 ∞ x 2 e − x 3 d x is convergent or not, and find its value if it is convergent.

Solution Summary: The author explains that the improper integral is convergent, and its value is 13.

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Whether the improper integral ∫0∞x2e−x3 dx is convergent or not, and find its

value if it is convergent.

Want to see the full answer?

Chapter 9 Solutions

Whether the improper integral ∫ 0 ∞ x 2 e − x 3 d x is convergent or not, and find its value if it is convergent.

Solution Summary: The author explains that the improper integral is convergent, and its value is 13.

Concept explainers

Videos

Whether the improper integral ∫0∞x2e−x3 dx is convergent or not, and find its value if it is convergent.

Want to see the full answer?

Chapter 9 Solutions

Whether the improper integral ∫0∞x2e−x3 dx is convergent or not, and find its

value if it is convergent.