Explanation Given: • Integral: ∫ sin n x 0 π 2 ⋅ cos x ⋅ d x • n ≥ 0 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 . U-Substitution is a technique we use when the integrand is a composite function. A composite function is a function inside another function of the form f ( u ( x ) ) where f ( x ) is the outside function and u ( x ) is the inside function. The first thing we will need to do is to recognize that we are being asked to integrate a product of a function and its derivative, and it takes the form of a composite function. This can be generalized for any product that is a composite function multiplied by the derivative of the inner function. Here in this problem we have to find the value of the given integral ∫ sin n x 0 π 2 ⋅ cos x ⋅ d x . To find the value of the given integral, we have to use substitution process. Here we can substitute u = sin x . Then the given integral becomes in terms of u . We have to integrate with respect to u . By applying the new limits of u which we found by calculation, we obtain the final solution. 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. ∫ sin n x 0 π 2 ⋅ cos x ⋅ d x for n ≥ 0 Step 2 : Write the value of u which we consider to substitute...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 5.7, Problem 85E

To determine

The value of the given integral for the provided value of $n \geq 0$

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Chapter 5.7, Problem 84E

Chapter 5.7, Problem 86E

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Chapter 5 Solutions

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