Explanation Given: • ∫ x n ⋅ ln x ⋅ d x for n ≠ 1 • Second Integration : ∫ x − 1 ⋅ ln x ⋅ d x Key Concepts Used: Integration by Parts is one of the fundamental techniques of integration. It is used for treating particular classes of functions such as trigonometric and rational functions. The Integration by Parts Formula: ∫ u ( x ) ⋅ v ' ( x ) ⋅ d x = u ( x ) ⋅ v ( x ) − ∫ u ' ( x ) ⋅ v ( x ) ⋅ d x From the above formula we come to know that it applies to a product u ( x ) ⋅ v ' ( x ) , we should consider using it when the integrand is a product of two functions. Here in this problem we have to prove the given equation of reduction formula. For that we use integration by parts. For this problem, in the given integration ∫ x n ⋅ ln x ⋅ d x we consider u ( x ) = ln x and v ' ( x ) = x n by means of which we have to find the value of given integral. Formula Used: The Integration by Parts Formula: ∫ u ( x ) ⋅ v ' ( x ) ⋅ d x = u ( x ) ⋅ v ( x ) − ∫ u ' ( x ) ⋅ v ( x ) ⋅ d x Calculation: Description: ∫ u ( x ) ⋅ v ' ( x ) ⋅ d x = u ( x ) ⋅ v ( x ) − ∫ u ' ( x ) ⋅ v ( x ) ⋅ d x Step 1 : Write the given integral. ∫ x n ⋅ e − x ⋅ d x Step 2 : For this problem, by comparing the given integration with the formula, write the values of u ( x ) a n d v ' ( x ) . C o m p a r i n g ∫ u ( x ) ⋅ v ' ( x ) ⋅ d x , a n d ∫ x n ⋅ ln x ⋅ d x w e f o u n d u ( x ) = ln x v ' ( x ) = x n Step 3 : Find the values of v ( x ) , a n d u ' ( x ) which are unknown values in the given formula. u ( x ) = ln x u ' ( x ) = d u d x = d d x ( ln x ) = 1 x v ' ( x ) = x n v ( x ) = ∫ v ' ( x ) ⋅ d x = ∫ x n ⋅ d x = x n + 1 n + 1 v ( x ) = x n + 1 n + 1 Step 4 : Substitute the values of u ( x ) , u ' ( x ) , v ( x ) , v ' ( x ) in the formula to derive the value of the given integral S u b s t i t u t e u ( x ) = ln x , u ' ( x ) = 1 x ,...

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.1, Problem 64E

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• The value of the given integral

• The method used to integrate the given second integral

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Chapter 8.1, Problem 63E

Chapter 8.1, Problem 65E

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Chapter 8 Solutions

Calculus - Standalone book

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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• The value of the given integral • The method used to integrate the given second integral

Solution Summary: The author explains that integration by parts is one of the fundamental techniques of integration, used for treating particular classes of functions such as trigonometric and rational functions.

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• The value of the given integral • The method used to integrate the given second integral

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Chapter 8 Solutions

• The value of the given integral

• The method used to integrate the given second integral