Explanation Given: The definite integral ∫ 1 3 2 x x 2 + 1 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. Power rule of integration: ∫ x n d x = x n + 1 n + 1 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 ∫ 2 x x 2 + 1 d x = ∫ 2 x ( x 2 + 1 ) 1 / 2 d x Substitute x 2 + 1 = u ⇒ 2 x d x = d u , so above integral becomes ∫ 2 x ( x 2 + 1 ) 1 / 2 d x = ∫ ( u ) 1 / 2 d u = u 1 2 + 1 1 2 + 1 = 2 u 3 / 2 3 substitute bave u = x 2 + 1 to get = 2 3 ( x 2 + 1 ) 3 / 2 So, the antiderivative of the function f ( x ) = 2 x x 2 + 1 is F ( x ) = 2 3 ( x 2 + 1 ) 3 / 2

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.5, Problem 26S

To determine

The exact average value of the definite integral ∫132x x 2+1 dx using the Fundamental Theorem of Calculus.

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Chapter 4.5, Problem 25S

Chapter 4.5, Problem 27S

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The exact average value of the definite integral ∫ 1 3 2 x x 2 + 1 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 ∫132x x 2+1 dx using the Fundamental Theorem of Calculus.

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