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87. The Integration by Parts formula can be written du = uV - where V (x) satisfies V'(x) = v(x). (a) Show directly that the right-hand side of Eq. (8) does not change if V (x) is replaced by V (x) + C, where C is a constant. (b) Use u = tan-1x and v = x in Eq. (8) to calculate | x tan-x dx, but carry out the calculation twice: first with V (x) = x² and then with V (x) = x² +. Which choice of V (x) results in a simpler calculation?

87. The Integration by Parts formula can be written du = uV - where V (x) satisfies V'(x) = v(x). (a) Show directly that the right-hand side of Eq. (8) does not change if V (x) is replaced by V (x) + C, where C is a constant. (b) Use u = tan-1x and v = x in Eq. (8) to calculate | x tan-x dx, but carry out the calculation twice: first with V (x) = x² and then with V (x) = x² +. Which choice of V (x) results in a simpler calculation?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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87. The Integration by Parts formula can be written du = uV - where V (x) satisfies V'(x) = v(x). (a) Show directly that the right-hand side of Eq. (8) does not change if V (x) is replaced by V (x) + C, where C is a constant. (b) Use u = tan-1x and v = x in Eq. (8) to calculate | x tan-x dx, but carry out the calculation twice: first with V (x) = x² and then with V (x) = x² +. Which choice of V (x) results in a simpler calculation?
Transcribed Image Text:87. The Integration by Parts formula can be written du = uV - where V (x) satisfies V'(x) = v(x). (a) Show directly that the right-hand side of Eq. (8) does not change if V (x) is replaced by V (x) + C, where C is a constant. (b) Use u = tan-1x and v = x in Eq. (8) to calculate | x tan-x dx, but carry out the calculation twice: first with V (x) = x² and then with V (x) = x² +. Which choice of V (x) results in a simpler calculation?
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