Explanation Given: • I = ∫ d θ 1 + tan 6000 θ 0 π 2 • J = ∫ d θ 1 + cot 6000 θ 0 π 2 • I + J = ∫ d θ 0 π 2 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. 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 integral I = ∫ d θ 1 + tan 6000 θ 0 π 2 is equal to the integral J = ∫ d θ 1 + cot 6000 θ 0 π 2 . Then we have to find the value of integral by using the equation I + J = ∫ d θ 0 π 2 .We have to substitute u = π 2 − θ in the integral I = ∫ d θ 1 + tan 6000 θ 0 π 2 to show that I = J . To obtain the value of the given integral through the equation, we can apply I + J = 2 I as I = J . Thus we can evaluate the value of the given integral. 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. I = ∫ d θ 1 + tan 6000 θ 0 π 2 Step 2 : Write the value of u which we consider to substitute. We consider u = π 2 − θ . 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. I = ∫ d θ 1 + tan 6000 θ 0 π 2 u = π 2 − θ ; d u d θ = d d θ ( π 2 − θ ) = 0 − 1 ⇒ d u = − d θ W h e n θ = 0 , u = ( π 2 − 0 ) = π 2 W h e n θ = π 2 , u = ( π 2 − π 2 ) = 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 to show that I = J ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5.7, Problem 91E

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The value of given integral $I$ by showing it equals the provided integral $J$ and evaluating through their sum

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Chapter 5.7, Problem 90E

Chapter 5.7, Problem 92E

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The value of given integral I by showing it equals the provided integral J and evaluating through their sum

Solution Summary: The author explains how to determine the value of given integral I by showing it equals the provided integral J and evaluating through their sum.

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The value of given integral $I$ by showing it equals the provided integral $J$ and evaluating through their sum

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The value of given integral I by showing it equals the provided integral J and evaluating through their sum

Solution Summary: The author explains how to determine the value of given integral I by showing it equals the provided integral J and evaluating through their sum.

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The value of given integral Iby showing it equals the provided integral J and evaluating through their sum

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The value of given integral $I$ by showing it equals the provided integral $J$ and evaluating through their sum