Explanation Given Information: The provided integral is, ∫ − ∞ ∞ 2 x ( x 2 + 1 ) 2 d x Formula used: According to the power rule of integrals, ∫ x n d x = x n + 1 n + 1 + C If the limit defining the improper integral is a unique finite number, then integral converges else it diverges, ∫ − ∞ ∞ f ( x ) d x = lim a → − ∞ ∫ a c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x Calculation: Consider the provided integral, ∫ − ∞ ∞ 2 x ( x 2 + 1 ) 2 d x Now, use the formula, ∫ − ∞ ∞ f ( x ) d x = lim a → − ∞ ∫ a c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x To rewrite the integral as, ∫ − ∞ ∞ 2 x ( x 2 + 1 ) 2 d x = lim a → − ∞ ∫ a 0 2 x ( x 2 + 1 ) 2 d x + lim b → ∞ ∫ 0 b 2 x ( x 2 + 1 ) 2 d x Let x 2 + 1 = t , then differentiate both sides with respect to x , 2 x d x = d t Thus, the integral becomes, ∫ − ∞ ∞ 2 x ( x 2 + 1 ) 2 d x = lim a → − ∞ ∫ a 0 2 x ( x 2 + 1 ) 2 d x + lim b → ∞ ∫ 0 b 2 x ( x 2 + 1 ) 2 d x = lim a → − ∞ ∫ a 0 d t t 2 + lim b → ∞ ∫ 0 b d t t 2 Now, use the power rule of integrals to evaluate the integral,

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Chapter 13.7, Problem 17E

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To calculate: The value of the improper integral ∫−∞∞2x(x2+1)2dx if it converges.

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Chapter 13.7, Problem 16E

Chapter 13.7, Problem 18E

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Chapter 13 Solutions

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The value of the improper integral ∫ − ∞ ∞ 2 x ( x 2 + 1 ) 2 d x if it converges.

Solution Summary: The author explains how to calculate the value of the provided integral, if it converges.

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To calculate: The value of the improper integral ∫−∞∞2x(x2+1)2dx if it converges.

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Chapter 13 Solutions