Explanation Given: The integral ∫ x n ln x d x and n = 0,1, 2, 3, and 4. Calculation: For n =0, the integral ∫ x n ln x d x becomes ∫ ln x d x . Integrating by parts and letting u = ln | x | , such that d u = 1 x d x ------(1) Also, we let d v = d x , such that v = x + C ------(2) ∫ ln x d x = x ln x − ∫ ( 1 x ) ( x ) d x ∫ ln x d x = x ln x − ∫ d x ∫ ln x d x = x ln x − x + C ------(2) Using a Computer Algebraic System, we can verify that the value of ∫ ln x d x is ----------(3) For n =1, the integral ∫ x n ln x d x becomes ∫ x ln x d x . Integrating by parts and letting u = ln | x | , such that d u = 1 x d x ------(4) Also, we let d v = x d x , such that v = x 2 2 + C ------(5) ∫ x ln x d x = x 2 2 ln x − ∫ ( 1 x ) ( x 2 2 ) d x ∫ x ln x d x = x 2 2 ln x − 1 2 ∫ x d x ∫ x ln x d x = x 2 2 ln x − x 2 4 + C ------(6) Using a Computer Algebraic System, we can verify that the value of ∫ x ln x d x is ---------(7) For n =2, the integral ∫ x n ln x d x becomes ∫ x 2 ln x d x . Integrating by parts and letting u = ln | x | , such that d u = 1 x d x ------(8) Also, we let d v = x 2 d x , such that v = x 3 3 + C ------(9) ∫ x 2 ln x d x = x 3 3 ln x − ∫ ( 1 x ) ( x 3 3 ) d x ∫ x 2 ln x d x = x 3 3 ln x − 1 3 ∫ x 2 d x ∫ x 2 ln x d x = x 3 3 ln x − x 3 9 + C ------(10) Using a Computer Algebraic System, we can verify that the value of ∫ x 2 ln x d x is -----------(11) For n =3, the integral ∫ x n ln x d x becomes ∫ x 3 ln x d x

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Quadratic Equation

When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations …

Demand and Supply Function

The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to …

Videos

Question

Chapter 8.2, Problem 65E

To determine

To calculate: ∫xnlnxdx using a Computer Algebra System and obtain a general rule.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.2, Problem 64E

Chapter 8.2, Problem 66E

Students have asked these similar questions

The correct answer is C,i know that we need to use stokes theorem and parametrize the equations then write the equation F with respect to the curve but i cant seem to find a way to do it, the integral should be from 0 to 2pi but i might be wrongcould you show me the steps to get to 18pi

A 10-ft boom is acted upon by the 810-lb force as shown in the figure. D 6 ft 6 ft E B 7 ft C 6 ft 4 ft W Determine the tension in each cable and the reaction at the ball-and-socket joint at A. The tension in cable BD is lb. The tension in cable BE is lb. The reaction at A is ( lb) i + Ib) j. (Include a minus sign if necessary.)

the correct answer is A could you show me why

Chapter 8 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Choosing an Antiderivative In Exercises 14, select...Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, selectthe...

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.

∫ x n ln x d x using a Computer Algebra System and obtain a general rule.

Solution Summary: The author explains how to calculate the integral displaystyle, in a Computer Algebra System, and obtain the general rule.

Concept explainers

Videos

To calculate: ∫xnlnxdx using a Computer Algebra System and obtain a general rule.

Want to see the full answer?

Chapter 8 Solutions