Explanation Given information: The definite integral is given as ∫ 0 3 x ( x 2 + 4 ) 2 d x and the change of variable is given as u = x 2 + 4 . Formula used: The fundamental theorem of calculus part(I). If F is an antiderivative of f , then, ∫ a b f ( x ) d x = F ( b ) − F ( a ) When a substitution is performed using differentials the change of variable formula is used. ∫ f ( u ( x ) ) u ' ( x ) d x = ∫ f ( u ) d u The power rule of indefinite integrals. ∫ x n d x = x n + 1 n + 1 + C when n ≠ ( − 1 ) The sum rule for indefinite integrals, ∫ ( f ( x ) + g ( x ) ) d x = ∫ f ( x ) d x + ∫ g ( x ) d x The multiple rule of antiderivatives. ∫ c f ( x ) d x = c ∫ f ( x ) d x Calculation: First let’s find expansion of the algebraic expression x ( x 2 + 4 ) 2 x ( x 2 + 4 ) 2 = x ( ( x 2 ) 2 + ( 2 × x 2 ( 4 ) ) + ( 4 ) 2 ) = x ( x 4 + 8 x 2 + 16 ) = x 5 + 8 x 3 + 16 x Then the integration can be done as, ∫ 0 3 x ( x 2 + 4 ) 2 d x = ∫ 0 3 ( x 5 + 8 x 3 + 16 x ) d x = ∫ 0 3 x 5 d x + ∫ 0 3 8 x 3 d x + ∫ 0 3 16 x d x = ∫ 0 3 x 5 d x + 8 ∫ 0 3 x 3 d x + 16 ∫ 0 3 x d x = [ x 6

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Author: Rogawski

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Chapter 5.7, Problem 72E

To determine

To find: The value of the definite integral ∫03x( x 2 +4)2dx by multiplying the function first and then by using the substitution.

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Chapter 5.7, Problem 71E

Chapter 5.7, Problem 73E

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The value of the definite integral ∫ 0 3 x ( x 2 + 4 ) 2 d x by multiplying the function first and then by using the substitution.

Solution Summary: The author explains how to find the value of the definite integral by multiplying the function first and then using the substitution.

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To find: The value of the definite integral ∫03x( x 2 +4)2dx by multiplying the function first and then by using the substitution.

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Chapter 5 Solutions