Explanation Given information: Given integral is ∫ ( x + 5 ) 2 d x The first way to be used is multiplying out and the second method is by substituting w = x + 5. Calculation: (i) Consider the integral ∫ ( x + 5 ) 2 d x Simplifying, ∫ ( x + 5 ) 2 d x = ∫ ( x + 5 ) ( x + 5 ) d x ∫ ( x + 5 ) 2 d x = ∫ ( x 2 + 10 x + 25 ) d x Distributing the integral and then simplifying, ∫ ( x + 5 ) 2 d x = ∫ x 2 d x + ∫ 10 x d x + ∫ 25 d x ∫ ( x + 5 ) 2 d x = x 2 + 1 2 + 1 + 10 x 1 + 1 1 + 1 + 25 x + C ∫ ( x + 5 ) 2 d x = x 3 3 + 10 x 2 2 + 25 x + C ∫ ( x + 5 ) 2 d x = x 3 3 + 5 x 2 + 25 x + C Where C is any constant To determine (b) To explain: To check if the solutions obtained in part (a) by using two methods are equal and to explain the reason for the same.

Applied Calculus, 6e WileyPLUS + Loose-leaf

6th Edition

ISBN: 9781119408901

Author: Deborah Hughes-Hallett, Andrew M. Gleason, Patti Frazer Lock, Daniel E. Flath

Publisher: Wiley (WileyPLUS Products)

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Chapter 6.6, Problem 65P

To determine

(a)

To calculate the given integral in two ways: (i) by multiplying out and (ii) by substitution.

To determine

(b)

To explain:

To check if the solutions obtained in part (a) by using two methods are equal and to explain the reason for the same.

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Chapter 6.6, Problem 64P

Chapter 6.6, Problem 66P

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Applied Calculus, 6e WileyPLUS + Loose-leaf

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(a) To calculate the given integral in two ways: (i) by multiplying out and (ii) by substitution.

Solution Summary: The author explains how to calculate the given integral by multiplying out and substituting w = x + 5.