Explanation Given: ∫ sec 2 x ⋅ tan 4 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: Exponent Rules Rule 1 Rule 2 Rule 3 Rule 4 a = a 1 2 a 3 = a 1 3 a m n = a m n 1 a m = a − m Rules used here in this problem are highlighted in green in the below table: Basic Properties/Rules/Formulas of Common Derivative ( f ( x ) ) ' notation indicates derivative of function Polynomials Function ( c ) ' = 0 , where c any constant ( x ) ' = d x ( c ⋅ x ) ' = c ⋅ d x ( x n ) ' = n ⋅ x n − 1 ⋅ d x ( c ⋅ x n ) ' = c ⋅ n ⋅ x n − 1 ⋅ d x Trigonometric Functions ( sin x ) ' = cos x ⋅ d x ( cos x ) ' = − sin x ⋅ d x ( tan x ) ' = sec 2 x ⋅ d x ( sec x ) ' = sec x ⋅ tan x ⋅ d x ( csc x ) ' = − csc x ⋅ cot x ⋅ d x ( cot x ) ' = − csc 2 x ⋅ d x Inverse Trigonometric Functions ( sin − 1 x ) ' = 1 1 − x 2 ⋅ d x ( cos − 1 x ) ' = − 1 1 − x 2 ⋅ d x ( tan − 1 x ) ' = 1 1 + x 2 ⋅ d x ( sec − 1 x ) ' = 1 | x | ⋅ x 2 − 1 ⋅ d x ( csc − 1

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5.7, Problem 56E

To determine

To evaluate:

The given indefinite integral

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Chapter 5.7, Problem 55E

Chapter 5.7, Problem 57E

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Chapter 5 Solutions

Calculus - Standalone book

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

Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - Prob. 49E Ch. 5.1 - Prob. 50E Ch. 5.1 - Prob. 51E Ch. 5.1 - Prob. 52E Ch. 5.1 - Prob. 53E Ch. 5.1 - Prob. 54E Ch. 5.1 - Prob. 55E Ch. 5.1 - 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Prob. 12E Ch. 5.2 - Prob. 13E Ch. 5.2 - Prob. 14E Ch. 5.2 - Prob. 15E Ch. 5.2 - Prob. 16E Ch. 5.2 - Prob. 17E Ch. 5.2 - Prob. 18E Ch. 5.2 - Prob. 19E Ch. 5.2 - Prob. 20E Ch. 5.2 - Prob. 21E Ch. 5.2 - Prob. 22E Ch. 5.2 - Prob. 23E Ch. 5.2 - Prob. 24E Ch. 5.2 - Prob. 25E Ch. 5.2 - Prob. 26E Ch. 5.2 - Prob. 27E Ch. 5.2 - Prob. 28E Ch. 5.2 - Prob. 29E Ch. 5.2 - Prob. 30E Ch. 5.2 - Prob. 31E Ch. 5.2 - Prob. 32E Ch. 5.2 - Prob. 33E Ch. 5.2 - Prob. 34E Ch. 5.2 - Prob. 35E Ch. 5.2 - Prob. 36E Ch. 5.2 - Prob. 37E Ch. 5.2 - Prob. 38E Ch. 5.2 - Prob. 39E Ch. 5.2 - Prob. 40E Ch. 5.2 - Prob. 41E Ch. 5.2 - Prob. 42E Ch. 5.2 - Prob. 43E Ch. 5.2 - Prob. 44E Ch. 5.2 - Prob. 45E Ch. 5.2 - Prob. 46E Ch. 5.2 - Prob. 47E Ch. 5.2 - Prob. 48E Ch. 5.2 - Prob. 49E Ch. 5.2 - Prob. 50E Ch. 5.2 - Prob. 51E Ch. 5.2 - Prob. 52E Ch. 5.2 - Prob. 53E Ch. 5.2 - Prob. 54E Ch. 5.2 - Prob. 55E Ch. 5.2 - Prob. 56E Ch. 5.2 - Prob. 57E Ch. 5.2 - Prob. 58E Ch. 5.2 - Prob. 59E Ch. 5.2 - Prob. 60E Ch. 5.2 - Prob. 61E Ch. 5.2 - 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Prob. 71E Ch. 5.3 - Prob. 72E Ch. 5.3 - Prob. 73E Ch. 5.3 - Prob. 74E Ch. 5.3 - Prob. 75E Ch. 5.3 - Prob. 76E Ch. 5.3 - Prob. 77E Ch. 5.3 - Prob. 78E Ch. 5.3 - Prob. 79E Ch. 5.3 - Prob. 80E Ch. 5.3 - Prob. 81E Ch. 5.3 - Prob. 82E Ch. 5.3 - Prob. 83E Ch. 5.3 - Prob. 84E Ch. 5.3 - Prob. 85E Ch. 5.3 - Prob. 86E Ch. 5.3 - Prob. 87E Ch. 5.3 - Prob. 88E Ch. 5.4 - Prob. 1PQ Ch. 5.4 - Prob. 2PQ Ch. 5.4 - Prob. 3PQ Ch. 5.4 - Prob. 4PQ Ch. 5.4 - Prob. 1E Ch. 5.4 - Prob. 2E Ch. 5.4 - Prob. 3E Ch. 5.4 - Prob. 4E Ch. 5.4 - Prob. 5E Ch. 5.4 - Prob. 6E Ch. 5.4 - Prob. 7E Ch. 5.4 - Prob. 8E Ch. 5.4 - Prob. 9E Ch. 5.4 - Prob. 10E Ch. 5.4 - Prob. 11E Ch. 5.4 - Prob. 12E Ch. 5.4 - Prob. 13E Ch. 5.4 - Prob. 14E Ch. 5.4 - Prob. 15E Ch. 5.4 - Prob. 16E Ch. 5.4 - Prob. 17E Ch. 5.4 - Prob. 18E Ch. 5.4 - Prob. 19E Ch. 5.4 - Prob. 20E Ch. 5.4 - Prob. 21E Ch. 5.4 - Prob. 22E Ch. 5.4 - Prob. 23E Ch. 5.4 - Prob. 24E Ch. 5.4 - Prob. 25E Ch. 5.4 - Prob. 26E Ch. 5.4 - Prob. 27E Ch. 5.4 - Prob. 28E Ch. 5.4 - Prob. 29E Ch. 5.4 - Prob. 30E Ch. 5.4 - Prob. 31E Ch. 5.4 - Prob. 32E Ch. 5.4 - Prob. 33E Ch. 5.4 - Prob. 34E Ch. 5.4 - Prob. 35E Ch. 5.4 - Prob. 36E Ch. 5.4 - Prob. 37E Ch. 5.4 - Prob. 38E Ch. 5.4 - Prob. 39E Ch. 5.4 - Prob. 40E Ch. 5.4 - Prob. 41E Ch. 5.4 - Prob. 42E Ch. 5.4 - Prob. 43E Ch. 5.4 - Prob. 44E Ch. 5.4 - Prob. 45E Ch. 5.4 - Prob. 46E Ch. 5.4 - Prob. 47E Ch. 5.4 - Prob. 48E Ch. 5.4 - Prob. 49E Ch. 5.4 - Prob. 50E Ch. 5.4 - Prob. 51E Ch. 5.4 - Prob. 52E Ch. 5.4 - Prob. 53E Ch. 5.4 - Prob. 54E Ch. 5.4 - Prob. 55E Ch. 5.4 - Prob. 56E Ch. 5.5 - Prob. 1PQ Ch. 5.5 - Prob. 2PQ Ch. 5.5 - Prob. 3PQ Ch. 5.5 - Prob. 4PQ Ch. 5.5 - Prob. 1E Ch. 5.5 - Prob. 2E Ch. 5.5 - Prob. 3E Ch. 5.5 - Prob. 4E Ch. 5.5 - Prob. 5E Ch. 5.5 - Prob. 6E Ch. 5.5 - Prob. 7E Ch. 5.5 - Prob. 8E Ch. 5.5 - Prob. 9E Ch. 5.5 - Prob. 10E Ch. 5.5 - Prob. 11E Ch. 5.5 - Prob. 12E Ch. 5.5 - Prob. 13E Ch. 5.5 - Prob. 14E Ch. 5.5 - Prob. 15E Ch. 5.5 - Prob. 16E Ch. 5.5 - Prob. 17E Ch. 5.5 - Prob. 18E Ch. 5.5 - Prob. 19E Ch. 5.5 - Prob. 20E Ch. 5.5 - Prob. 21E Ch. 5.5 - Prob. 22E Ch. 5.5 - Prob. 23E Ch. 5.5 - Prob. 24E Ch. 5.5 - Prob. 25E Ch. 5.5 - Prob. 26E Ch. 5.5 - Prob. 27E Ch. 5.5 - Prob. 28E Ch. 5.5 - Prob. 29E Ch. 5.5 - Prob. 30E Ch. 5.5 - Prob. 31E Ch. 5.5 - Prob. 32E Ch. 5.5 - Prob. 33E Ch. 5.5 - Prob. 34E Ch. 5.5 - Prob. 35E Ch. 5.5 - Prob. 36E Ch. 5.5 - Prob. 37E Ch. 5.5 - Prob. 38E Ch. 5.5 - Prob. 39E Ch. 5.5 - Prob. 40E Ch. 5.5 - Prob. 41E Ch. 5.5 - Prob. 42E Ch. 5.5 - Prob. 43E Ch. 5.5 - Prob. 44E Ch. 5.5 - Prob. 45E Ch. 5.5 - Prob. 46E Ch. 5.5 - Prob. 47E Ch. 5.5 - Prob. 48E Ch. 5.5 - Prob. 49E Ch. 5.5 - Prob. 50E Ch. 5.5 - Prob. 51E Ch. 5.5 - Prob. 52E Ch. 5.5 - Prob. 53E Ch. 5.6 - Prob. 1PQ Ch. 5.6 - Prob. 2PQ Ch. 5.6 - Prob. 3PQ Ch. 5.6 - Prob. 1E Ch. 5.6 - Prob. 2E Ch. 5.6 - Prob. 3E Ch. 5.6 - Prob. 4E Ch. 5.6 - Prob. 5E Ch. 5.6 - Prob. 6E Ch. 5.6 - Prob. 7E Ch. 5.6 - Prob. 8E Ch. 5.6 - Prob. 9E Ch. 5.6 - Prob. 10E Ch. 5.6 - Prob. 11E Ch. 5.6 - Prob. 12E Ch. 5.6 - Prob. 13E Ch. 5.6 - Prob. 14E Ch. 5.6 - Prob. 15E Ch. 5.6 - Prob. 16E Ch. 5.6 - Prob. 17E Ch. 5.6 - Prob. 18E Ch. 5.6 - Prob. 19E Ch. 5.6 - Prob. 20E Ch. 5.6 - Prob. 21E Ch. 5.6 - Prob. 22E Ch. 5.6 - Prob. 23E Ch. 5.6 - Prob. 24E Ch. 5.6 - Prob. 25E Ch. 5.6 - Prob. 26E Ch. 5.6 - Prob. 27E Ch. 5.6 - Prob. 28E Ch. 5.6 - Prob. 29E Ch. 5.6 - Prob. 30E Ch. 5.6 - Prob. 31E Ch. 5.7 - Prob. 1PQ Ch. 5.7 - Prob. 2PQ Ch. 5.7 - Prob. 3PQ Ch. 5.7 - Prob. 1E Ch. 5.7 - Prob. 2E Ch. 5.7 - Prob. 3E Ch. 5.7 - Prob. 4E Ch. 5.7 - Prob. 5E Ch. 5.7 - Prob. 6E Ch. 5.7 - Prob. 7E Ch. 5.7 - Prob. 8E Ch. 5.7 - Prob. 9E Ch. 5.7 - Prob. 10E Ch. 5.7 - Prob. 11E Ch. 5.7 - Prob. 12E Ch. 5.7 - Prob. 13E Ch. 5.7 - Prob. 14E Ch. 5.7 - Prob. 15E Ch. 5.7 - Prob. 16E Ch. 5.7 - Prob. 17E Ch. 5.7 - Prob. 18E Ch. 5.7 - Prob. 19E Ch. 5.7 - Prob. 20E Ch. 5.7 - Prob. 21E Ch. 5.7 - Prob. 22E Ch. 5.7 - Prob. 23E Ch. 5.7 - Prob. 24E Ch. 5.7 - Prob. 25E Ch. 5.7 - Prob. 26E Ch. 5.7 - Prob. 27E Ch. 5.7 - Prob. 28E Ch. 5.7 - Prob. 29E Ch. 5.7 - Prob. 30E Ch. 5.7 - Prob. 31E Ch. 5.7 - Prob. 32E Ch. 5.7 - Prob. 33E Ch. 5.7 - Prob. 34E Ch. 5.7 - Prob. 35E Ch. 5.7 - Prob. 36E Ch. 5.7 - Prob. 37E Ch. 5.7 - Prob. 38E Ch. 5.7 - Prob. 39E Ch. 5.7 - Prob. 40E Ch. 5.7 - Prob. 41E Ch. 5.7 - Prob. 42E Ch. 5.7 - Prob. 43E Ch. 5.7 - Prob. 44E Ch. 5.7 - Prob. 45E Ch. 5.7 - Prob. 46E Ch. 5.7 - Prob. 47E Ch. 5.7 - Prob. 48E Ch. 5.7 - Prob. 49E Ch. 5.7 - Prob. 50E Ch. 5.7 - Prob. 51E Ch. 5.7 - Prob. 52E Ch. 5.7 - Prob. 53E Ch. 5.7 - Prob. 54E Ch. 5.7 - Prob. 55E Ch. 5.7 - Prob. 56E Ch. 5.7 - Prob. 57E Ch. 5.7 - Prob. 58E Ch. 5.7 - Prob. 59E Ch. 5.7 - Prob. 60E Ch. 5.7 - Prob. 61E Ch. 5.7 - Prob. 62E Ch. 5.7 - Prob. 63E Ch. 5.7 - Prob. 64E Ch. 5.7 - Prob. 65E Ch. 5.7 - Prob. 66E Ch. 5.7 - Prob. 67E Ch. 5.7 - Prob. 68E Ch. 5.7 - Prob. 69E Ch. 5.7 - Prob. 70E Ch. 5.7 - Prob. 71E Ch. 5.7 - Prob. 72E Ch. 5.7 - Prob. 73E Ch. 5.7 - Prob. 74E Ch. 5.7 - Prob. 75E Ch. 5.7 - Prob. 76E Ch. 5.7 - Prob. 77E Ch. 5.7 - Prob. 78E Ch. 5.7 - Prob. 79E Ch. 5.7 - Prob. 80E Ch. 5.7 - Prob. 81E Ch. 5.7 - Prob. 82E Ch. 5.7 - Prob. 83E Ch. 5.7 - Prob. 84E Ch. 5.7 - Prob. 85E Ch. 5.7 - Prob. 86E Ch. 5.7 - Prob. 87E Ch. 5.7 - Prob. 88E Ch. 5.7 - Prob. 89E Ch. 5.7 - Prob. 90E Ch. 5.7 - Prob. 91E Ch. 5.7 - Prob. 92E Ch. 5.7 - Prob. 93E Ch. 5.7 - Prob. 94E Ch. 5.7 - Prob. 95E Ch. 5 - Prob. 1CRE Ch. 5 - Prob. 2CRE Ch. 5 - Prob. 3CRE Ch. 5 - Prob. 4CRE Ch. 5 - Prob. 5CRE Ch. 5 - Prob. 6CRE Ch. 5 - Prob. 7CRE Ch. 5 - Prob. 8CRE Ch. 5 - Prob. 9CRE Ch. 5 - Prob. 10CRE Ch. 5 - Prob. 11CRE Ch. 5 - Prob. 12CRE Ch. 5 - Prob. 13CRE Ch. 5 - Prob. 14CRE Ch. 5 - Prob. 15CRE Ch. 5 - Prob. 16CRE Ch. 5 - Prob. 17CRE Ch. 5 - Prob. 18CRE Ch. 5 - Prob. 19CRE Ch. 5 - Prob. 20CRE Ch. 5 - Prob. 21CRE Ch. 5 - Prob. 22CRE Ch. 5 - Prob. 23CRE Ch. 5 - Prob. 24CRE Ch. 5 - Prob. 25CRE Ch. 5 - Prob. 26CRE Ch. 5 - Prob. 27CRE Ch. 5 - Prob. 28CRE Ch. 5 - Prob. 29CRE Ch. 5 - Prob. 30CRE Ch. 5 - Prob. 31CRE Ch. 5 - Prob. 32CRE Ch. 5 - Prob. 33CRE Ch. 5 - Prob. 34CRE Ch. 5 - Prob. 35CRE Ch. 5 - Prob. 36CRE Ch. 5 - Prob. 37CRE Ch. 5 - Prob. 38CRE Ch. 5 - Prob. 39CRE Ch. 5 - Prob. 40CRE Ch. 5 - Prob. 41CRE Ch. 5 - Prob. 42CRE Ch. 5 - Prob. 43CRE Ch. 5 - Prob. 44CRE Ch. 5 - Prob. 45CRE Ch. 5 - Prob. 46CRE Ch. 5 - Prob. 47CRE Ch. 5 - Prob. 48CRE Ch. 5 - Prob. 49CRE Ch. 5 - Prob. 50CRE Ch. 5 - Prob. 51CRE Ch. 5 - Prob. 52CRE Ch. 5 - Prob. 53CRE Ch. 5 - Prob. 54CRE Ch. 5 - Prob. 55CRE Ch. 5 - Prob. 56CRE Ch. 5 - Prob. 57CRE Ch. 5 - Prob. 58CRE Ch. 5 - Prob. 59CRE Ch. 5 - Prob. 60CRE Ch. 5 - Prob. 61CRE Ch. 5 - Prob. 62CRE Ch. 5 - Prob. 63CRE Ch. 5 - Prob. 64CRE Ch. 5 - Prob. 65CRE Ch. 5 - Prob. 66CRE Ch. 5 - Prob. 67CRE Ch. 5 - Prob. 68CRE Ch. 5 - Prob. 69CRE Ch. 5 - Prob. 70CRE Ch. 5 - Prob. 71CRE Ch. 5 - Prob. 72CRE Ch. 5 - Prob. 73CRE Ch. 5 - Prob. 74CRE Ch. 5 - Prob. 75CRE Ch. 5 - Prob. 76CRE Ch. 5 - Prob. 77CRE Ch. 5 - Prob. 78CRE Ch. 5 - Prob. 79CRE Ch. 5 - Prob. 80CRE Ch. 5 - Prob. 81CRE Ch. 5 - Prob. 82CRE Ch. 5 - Prob. 83CRE Ch. 5 - Prob. 84CRE Ch. 5 - Prob. 85CRE Ch. 5 - Prob. 86CRE Ch. 5 - Prob. 87CRE Ch. 5 - Prob. 88CRE Ch. 5 - Prob. 89CRE Ch. 5 - Prob. 90CRE Ch. 5 - Prob. 91CRE Ch. 5 - Prob. 92CRE Ch. 5 - Prob. 93CRE Ch. 5 - Prob. 94CRE Ch. 5 - Prob. 95CRE

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The given indefinite 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 indefinite integral

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Chapter 5 Solutions

To evaluate:

The given indefinite integral