Explanation Given information: The improper integral is ∫ 0 ∞ x e − k x d x and lim b → ∞ b e − k b = 0 where k is a positive number. Formula used: If a is 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 . Integration by parts rule: If f ( x ) and g ( x ) are any two functions, and G ( x ) is an antiderivative of g ( x ) . Then, ∫ f ( x ) g ( x ) d x = f ( x ) G ( x ) − ∫ f ′ ( x ) G ( x ) d x . Calculation: Let b > 1 , and solve the improper integral by using the integration by parts rule. Hence f ( x ) = x and g ( x ) = e − k x , then f ′ ( x ) = 1 and G ( x ) = ∫ e − k x d x = ( − e − k x k ) . ∫ 0 b x e − k x d x = [ x ( − e − k x k ) | 0 b ] − ∫ 0 b ( − e − k x k ) d x = [ − x ( e − k x k ) | 0 b ] + ∫ 0 b e − k x k d x = [ − 1 k ( x e − k x ) | 0 b ] + ∫ 0 b ( e − k x k ) d x = [ − 1 k ( ( b ) e − k ( b ) − ( 0 ) e − k ( 0 ) ) ] + 1 k ∫ 0 b e − k x d x = [ − 1 k ( b e − k b − 0 ) ] + 1 k ( e − k x − k | 0 b ) = [ − b e − k b k ] − 1 k 2 ( e − k x | 0

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Chapter 9, Problem 54RE

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To calculate: The value of the improper integral ∫0∞xe−kx dx by using limb→∞ be−kb=0 where k is

a positive number.

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Chapter 9, Problem 53RE

Chapter 9, Problem 55RE

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The value of the improper integral ∫ 0 ∞ x e − k x d x by using lim b → ∞ b e − k b = 0 where k is a positive number.

Solution Summary: The author calculates the value of the improper integral by using undersetbto

Videos

To calculate: The value of the improper integral ∫0∞xe−kx dx by using limb→∞ be−kb=0 where k is

a positive number.

Want to see the full answer?

Chapter 9 Solutions

The value of the improper integral ∫ 0 ∞ x e − k x d x by using lim b → ∞ b e − k b = 0 where k is a positive number.

Solution Summary: The author calculates the value of the improper integral by using undersetbto

Videos

To calculate: The value of the improper integral ∫0∞xe−kx dx by using limb→∞ be−kb=0 where k is a positive number.

Want to see the full answer?

Chapter 9 Solutions

To calculate: The value of the improper integral ∫0∞xe−kx dx by using limb→∞ be−kb=0 where k is

a positive number.