Explanation Given information: The improper integral is ∫ 1 ∞ x + 2 x 2 + 4 x − 2 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 > 1 , solve the improper integral by using the substitution method. Let u = x 2 + 4 x − 2 . Differentiate with respect to x . ⇒ d u d x = 2 x + 4 ⇒ d u d x = 2 ( x + 2 ) ⇒ d u 2 = ( x + 2 ) d x Apply the substitution to the limits of integration. When x = 1 , then u = ( 1 ) 2 + 4 ( 1 ) − 2 = 3 ; and when x = b then u = b 2 + 4 b − 2 Therefore, the integral becomes, ∫ 1 b x + 2 x 2 + 4 x − 2 d x = ∫ 3 b 2 + 4 b − 2 1 u

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Chapter 9, Problem 49RE

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Whether the improper integral ∫1∞x+2x2+4x−2 dx is convergent or not, and find its value if it is convergent.

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Chapter 9, Problem 48RE

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Whether the improper integral ∫ 1 ∞ x + 2 x 2 + 4 x − 2 d x is convergent or not, and find its value if it is convergent.

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Whether the improper integral ∫1∞x+2x2+4x−2 dx is convergent or not, and find its value if it is convergent.

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