Explanation Given Information: The provided definite integral is ∫ 0 8 x x + 1 d x . Formula used: The integration by parts ∫ u d v = u v − ∫ v d u . Where, u and v are function of x . ∫ ( a x + b ) n d x = 1 a ( a x + b ) n + 1 n + 1 , n ≠ − 1 Calculation: Consider the definite integral ∫ 0 8 x x + 1 d x The above definite integral can be written as, ∫ 0 8 x x + 1 d x = ∫ 0 8 x ( x + 1 ) 1 2 d x Here, d v = ( x + 1 ) 1 2 d x and u = x First find v , d v = ( x + 1 ) 1 2 d x ∫ d v = ∫ ( x + 1 ) 1 2 d x On further solving, v = 2 3 ( x + 1 ) 3 2 …...…... (1) Find d u : u = x Differentiate both side with respect x ; d u d x = d ( x ) d x d u d x = 1 And, d u = 1 ⋅ d x …...…... (2) Apply integration by parts formula and substitute equation (1) and (2) in ∫ u d v = u v − ∫ v d u , ∫ x x + 1 d x = 2 3 x ( x + 1 ) 3 2 − 2 3 ∫ ( x + 1 ) 3 2 d x Now, take the lower limit as 0 and upper limit as 8 , ∫ 0 8 x x + 1 d x = [ 2 3 x ( x + 1 ) 3 2

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.1, Problem 43E

To determine

To calculate: The value of definite integral ∫08xx+1dx.

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Chapter 6 Solutions

Calculus: An Applied Approach (MindTap Course List)

Ch. 6.1 - Prob. 1CP Ch. 6.1 - Prob. 2CP Ch. 6.1 - Prob. 3CP Ch. 6.1 - Prob. 4CP Ch. 6.1 - Prob. 5CP Ch. 6.1 - Prob. 6CP Ch. 6.1 - Prob. 7CP Ch. 6.1 - Prob. 1SWU Ch. 6.1 - Prob. 2SWU Ch. 6.1 - Prob. 3SWU

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The value of definite integral ∫ 0 8 x x + 1 d x .

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To calculate: The value of definite integral ∫08xx+1dx.

Chapter 6 Solutions