Explanation Given: • ∫ f ( x − a a ) ⋅ d x 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 . Here in this problem we have to show that the value of the given integral ∫ f ( x − a a ) ⋅ d x equals zero if the function f is odd. To prove that, first we consider that the given function f is continuous. We know that if f ( x ) is an odd function, then f ( − x ) = − f ( x ) . By using this property of odd functions, we have to prove that the value of the given integral equals zero. 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. ∫ f ( x − a a ) ⋅ d x Step 2 : Write the value of u which we consider to substitute. We consider u = − x . Step 3 : Find the value of d u for the considered value of u and also check the limits of u with the given values of limit. ∫ f ( x − a a ) ⋅ d x u = − x ; d u d x = d d x ( − x ) = − 1 ⇒ d u = − d x W h e n x = − a , u = ( − ( − a ) ) = a ⇒ u ( − a ) = a W h e n x = 0 , u = ( 0 ) = 0 ⇒ u ( 0 ) = 0 Step 4 : Substitute the values of u a n d d u and its limits obtained from the above step in the given integral...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5.7, Problem 92E

To determine

To Prove:

That the given integral is zero if the function $f$ in the integral is odd by using substitution process

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Chapter 5.7, Problem 91E

Chapter 5.7, Problem 93E

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To Prove: That the given integral is zero if the function f in the integral is odd by using substitution process

Solution Summary: The author explains that the definite integral of f (x) over [a, b] is denoted by the integral sign.

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To Prove:

That the given integral is zero if the function $f$ in the integral is odd by using substitution process

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Chapter 5 Solutions