Explanation Given: ∫ 0 π / 4 tan 5 x ⋅ d x Key Concept Applied: 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 ( g ( x ) ) where f ( x ) is the outside function and g ( 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. This integration rule follows from the chain rule for differentiation , because U-Substitution is nothing more than the reverse of the chain rule. Substitution Rule To integrate: ∫ f ( g ( x ) ) ⋅ g ' ( x ) ⋅ d x ↗ ↖ I n s i d e f u n c t i o n O u t s i d e F u n c t i o n ( D e r i v a t i v e o f I n s i d e ) The essence of the substitution rule is to take an integral in terms of x and transform or change it into terms of u by finding the outside function and inside function, where the outside function is the derivative of the inside function. Then we will make a suitable substitution that will simplify our integrand so that we can integrate, as illustrated in three easy steps below: ∫ f ( g ( x ) ) ⋅ g ' ( x ) ⋅ d x , I f u = g ( x ) ⇒ d u = g ' ( x ) ⋅ d x , T h e n d x = d u g ' ( x ) A f t e r U − S u b t i t u t i o n , I n t e g r a l w i l l b e c o m e ... ∫ f ( u ) ⋅ g ' ( x ) ⋅ d u g ' ( x ) C a n c e l l i n g g ' ( x ) , w e g e t ... ∫ f ( u ) ⋅ d u Formula used: Common Integration Rules Trigonometric Identity ∫ ( f ( u ) ± g ( u ) ) ⋅ d u = ∫ f ( u ) ⋅ d u ± ∫ g ( u ) ⋅ d u ∫ u n d x = u n + 1 n + 1 + c , w h e r e n ≠ − 1 ∫ c f ( u ) ⋅ d u = c ⋅ ∫ f ( u ) ⋅ d u ∫ 1 u ⋅ d u = ln | u | + c tan 2 x = ( sec 2 x − 1 ) Calculation: Using Trigonometric Identity to rewrite the given integral ∫ 0 π / 4 tan 5 x ⋅ d x ∫ 0 π / 4 tan 5 x ⋅ d x = ∫ 0 π / 4 tan 4 x ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.2, Problem 58E

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To evaluate:

The given definite integral

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Chapter 8.2, Problem 57E

Chapter 8.2, Problem 59E

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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 given definite integral

Solution Summary: The author explains how U-Substitution is used when the integrand is a composite function. The essence of the substitution rule is to take an integral and transform it into terms of u.

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To evaluate: The given definite integral

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Chapter 8 Solutions

To evaluate:

The given definite integral