Explanation Given: • Substitution: u = 1 + x 1 n • Equation: ∫ 1 + x 1 n ⋅ d x = n ∫ u 1 2 ⋅ ( u − 1 ) n − 1 ⋅ d u • For n = 2 , 3 Key Concepts Used: Definite Integral: The definite integral of f ( x ) over [ a , b ] is denoted by the integral sign ∫ f ( x ) ⋅ d x a b . It is often called the integral of f ( x ) over [ a , b ] . The process of computing integrals is called Integration. The function f ( x ) is called the integrand. The endpoints a a n d b of [ a , b ] are called the Limits of Integration. Substitution Method: When the integrand has the form f ( u ( x ) ) ⋅ u ' ( x ) , we can use the equation d u = u ' ( x ) ⋅ d x to rewrite the entire integral (including the d x term) in terms of u and it’s differential d u as follows: ∫ f ( u ( x ) ) ⋅ u ' ( x ) ⋅ d x = ∫ f ( u ) ⋅ d u . This equation is called the Change of Variables Formula. It transforms an integral in the variable x into a simpler integral in the new variable u .Here in this problem we have to show that the given equation ∫ 1 + x 1 n ⋅ d x = n ∫ u 1 2 ⋅ ( u − 1 ) n − 1 ⋅ d u is true by using the given substitution u = 1 + x 1 n and also we have to find the value of the given integral for n = 2 , 3 . Here we have to find the valueof d u . If we substitute the obtained values in the integral ∫ 1 + x 1 n ⋅ d x and obtain the other part of the equation then the equation is true. Then we have to substitute n = 2 , 3 to find the corresponding integral values. Calculation: Description: ∫ f ( u ( x ) ) ⋅ u ' ( x ) ⋅ d x = ∫ f ( u ) ⋅ d u Step 1 :Write the provided integral for which we have to find the value. ∫ 1 + x 1 n ⋅ d x Step 2 : Write thevalue of u which we have to substitute. u = 1 + x 1 n Step 3 : Find the value of d u for the considered value of u . ∫ 1 + x 1 n ⋅ d x u = 1 + x 1 n ⇒ x 1 n = u − 1 T a k i n g n t h p o w e r o n b o t h s i d e s , ( x 1 n ) n = ( u − 1 ) n ⇒ x = ( u − 1 ) n d x d u = d d x ( u − 1 ) n = n ⋅ ( u − 1 ) n − 1 ⇒ d x = n ⋅ ( u − 1 ) n − 1 ⋅ d u Step 4 : Substitute the values obtained from the above step in the given integral to check whether the given integral satisfies the provided equivalent integral. S u b s t i t u t e x = ( u − 1 ) n , d x = n ⋅ ( u − 1 ) n − 1 ⋅ d u , ∫ 1 + x 1 n ⋅ d x = ∫ 1 + ( ( u − 1 ) n ) 1 n ⋅ n ⋅ ( u − 1 ) n − 1 ⋅ d u = ∫ 1 + u − 1 n ⋅ ( u − 1 ) n − 1 ⋅ d u = ∫ u 1 2 ⋅ n ⋅ ( u − 1 ) n − 1 ⋅ d u ∫ 1 + x 1 n ⋅ d x = n ∫ u 1 2 ⋅ ( u − 1 ) n − 1 ⋅ d u Hence the given equation is true is shown...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5.7, Problem 90E

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The value of given integral for expected values of $n$ by showing the given equation using the provided substitution is true

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Chapter 5.7, Problem 89E

Chapter 5.7, Problem 91E

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Chapter 5 Solutions

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Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

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The value of given integral for expected values of n by showing the given equation using the provided substitution is true

Solution Summary: The author explains how to determine the value of an integral for expected values of n by showing the given equation using the provided substitution.

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The value of given integral for expected values of $n$ by showing the given equation using the provided substitution is true

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The value of given integral for expected values of n by showing the given equation using the provided substitution is true

Solution Summary: The author explains how to determine the value of an integral for expected values of n by showing the given equation using the provided substitution.

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The value of given integral for expected values of nby showing the given equation using the provided substitution is true

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Chapter 5 Solutions

The value of given integral for expected values of $n$ by showing the given equation using the provided substitution is true