Explanation Given: ∫ d x ( x 2 + 25 ) 2 Key concepts 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: tan 2 u + 1 = sec 2 u cos 2 u = cos 2 u + 1 2 sin 2 u = 2 ⋅ sin u ⋅ cos u I n a R i g h t − a n g l e T r i a n g l e tan u = O p p o s i t e s i d e o f a n g l e u A d j a c e n t s i d e o f a n g l e u sin u = O p p o s i t e s i d e o f a n g l e u H y p o t e n u s e s i d e cos u = A d j a c e n t s i d e o f a n g l e u H y p o t e n u s e s i d e P y t h o g o r e a n T h e o r e m : a 2 + b 2 = c 2 I f x = tan u ; d x = sec 2 u ⋅ d u Calculation: Steps ∫ d x ( x 2 + 25 ) 2 Step 1: Use trigonometric substitution in order to rewrite the given integral in terms of u . x = 5 tan u x 2 = ( 5 tan u ) 2 = 25 tan 2 u x 2 + 25 = 25 tan 2 u + 25 x 2 + 25 = 25 ( tan 2 u + 1 ) ( x 2 + 25 ) 2 = 625 ( tan 2 u + 1 ) 2 d x = 5 ⋅ sec 2 u ⋅ d u ∫ d x ( x 2 + 25 ) 2 = ∫ 5 ⋅ sec 2 u ⋅ d u 625 ( tan 2 u + 1 ) 2 = 1 125 ∫ sec 2 u ⋅ d u ( tan 2 u + 1 ) 2 Step 2: Use Pythagorean identity to simplify the integral...

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 8, Problem 44CRE

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Chapter 8 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

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The given integral

Solution Summary: The author explains that U-Substitution is a technique used when the integrand is the composite function. The integration rule follows from the chain rule for differentiation.

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To evaluate: The given integral

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Chapter 8 Solutions

To evaluate:

The given integral