Explanation Given: The definite integral ∫ 0 1 / 2 1 1 + 4 x 2 d x . Concept Used: Fundamental Theorem of Calculus stats that if F ( x ) is the antiderivative of f ( x ) , then ∫ a b f ( x ) d x = F ( b ) − F ( a ) . The area that is bounded by the graph of f ( x ) above the x -axis is positive. The area that is bounded by the graph of f ( x ) below the x -axis is negative. ∫ 1 1 + ( a x ) 2 d x = 1 a tan − 1 ( a x ) Calculation: In order to find the exact value of the given definite integral first find the antiderivative of the function and then use the Fundamental theorem of calculus to simplify further. So, using the power rule of integration, the antiderivative of the given function is ∫ 1 1 + 4 x 2 d x = ∫ 1 1 + ( 2 x ) 2 d x = 1 2 tan − 1 ( 2 x ) So, the antiderivative of the function f ( x ) = 1 1 + 4 x 2 is F ( x ) = 1 2 tan − 1 ( 2 x )

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 4.5, Problem 30S

To determine

The exact average value of the definite integral ∫01/211+4 x 2 dx using the Fundamental Theorem of Calculus.

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Chapter 4.5, Problem 29S

Chapter 4.5, Problem 31S

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The exact average value of the definite integral ∫ 0 1 / 2 1 1 + 4 x 2 d x using the Fundamental Theorem of Calculus.

Solution Summary: The author explains the fundamental theorem of calculus and the power rule of integration to find the exact value of the given definite integral.

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The exact average value of the definite integral ∫01/211+4 x 2 dx using the Fundamental Theorem of Calculus.

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