Explanation Given: ∫ 0 1 ln ( 4 − 2 x ) d x Key Concept Applied: Integration by parts is a method for evaluating a difficult integral. When the integral is a product of functions, the integration by parts formula moves the products out the equation so the integral can be solved more easily. ∫ u ⋅ d v = u v − ∫ v ⋅ 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 ( 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: Integration by parts ∫ u ⋅ d v = u v − ∫ v ⋅ d u Formulas of Common Derivative ( c ) ' = 0 , where c any constant ( c ⋅ x ) ' = c ⋅ d x Formulas of Common Integral ∫ 1 u ⋅ d u = ln | u | + c ∫ ( f ( u ) ± g ( u ) ) ⋅ d u = ∫ f ( u ) ⋅ d u ± ∫ g ( u ) ⋅ d u ∫ a d x = a x ∫ c f ( u ) ⋅ d u = c ⋅ ∫ f ( u ) ⋅ d u Calculation: Description ∫ 0 1 ln ( 4 − 2 x ) d x Step 1: Set that inside function equal to u & v ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Videos

Question

Chapter 8, Problem 21CRE

To determine

To evaluate:

The given definite integral

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8, Problem 20CRE

Chapter 8, Problem 22CRE

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 definite integral

Solution Summary: The author explains the integration by parts formula for evaluating a difficult integral. U-Substitution is the reverse of the chain rule for differentiation.

Videos

To evaluate: The given definite integral

Want to see the full answer?

Chapter 8 Solutions

To evaluate:

The given definite integral