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...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Videos

Question

Chapter 8, Problem 44CRE

To determine

To evaluate:

The given integral

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8, Problem 43CRE

Chapter 8, Problem 45CRE

Students have asked these similar questions

a -> f(x) = f(x) = [x] show that whether f is continuous function or not(by using theorem) Muslim_maths

Use Green's Theorem to evaluate F. dr, where F = (√+4y, 2x + √√) and C consists of the arc of the curve y = 4x - x² from (0,0) to (4,0) and the line segment from (4,0) to (0,0).

Evaluate F. dr where F(x, y, z) = (2yz cos(xyz), 2xzcos(xyz), 2xy cos(xyz)) and C is the line π 1 1 segment starting at the point (8, ' and ending at the point (3, 2 3'6

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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.

Concept explainers

Videos

To evaluate: The given integral

Want to see the full answer?

Chapter 8 Solutions

To evaluate:

The given integral