How To Integrate These?

  • Thread starter basty
  • Start date
  • #1
95
0

Homework Statement



FIRST
##\int x^2 \ln x \ dx##

SECOND
##\int \frac{(x+1)^2}{x}dx##

THIRD
##\int \frac{x-2}{x+3}dx##

Homework Equations



N/A

The Attempt at a Solution


I tried using the integration by part method but it doesn't work.

For the FIRST problem:
Setting ##u=x^2## or ##u=\ln x## is not solving the problem.

For the SECOND problem:
Setting ##u=x+1## or ##u=x## is not solving the problem.

For the THIRD problem:
Setting ##u=x-2## or ##u=x+3## is not solving the problem.
 
Last edited:

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
17,240
7,057
You need to provide more details than this on your attempted solution. Show us what you did and what you got.
 
  • #3
95
0
You need to provide more details than this on your attempted solution. Show us what you did and what you got.

For the FIRST problem:
Setting ##u=x^2## or ##u=\ln x## is not solving the problem.

For the SECOND problem:
Setting ##u=x+1## or ##u=x## is not solving the problem.

For the THIRD problem:
Setting ##u=x-2## or ##u=x+3## is not solving the problem.
 
  • #4
ehild
Homework Helper
15,543
1,913
What is u? show how did you proceed in the first case.
How do you do the integration by parts?
 
  • #5
95
0
Give me some hints.
 
  • #6
pasmith
Homework Helper
2,036
661

Homework Statement



FIRST
##\int x^2 \ln x \ dx##

SECOND
##\int \frac{(x+1)^2}{x}dx##

THIRD
##\int \frac{x-2}{x+3}dx##

Homework Equations



N/A

The Attempt at a Solution


I tried using the integration by part method but it doesn't work.

For the FIRST problem:
Setting ##u=x^2## or ##u=\ln x## is not solving the problem.

Differentiating [itex]\ln x[/itex] and integrating [itex]x^2[/itex] is the correct method. What did you get when you tried it?

The remaining two problems do not require integration by parts. They can be done by direct integration after some algebraic manipulation of the integrand.
 
  • #7
95
0
Differentiating [itex]\ln x[/itex] and integrating [itex]x^2[/itex] is the correct method. What did you get when you tried it?

Thank you

This is the FIRST problem's solution done by me after getting a hint from you:
let ##u=\ln x## then ##\frac{du}{dx}=\frac{1}{x}## or ##du=\frac{1}{x}dx##
let ##dv=x^2## then ##v=\frac{1}{3}x^3##

So ##\int x^2 \ln x \ dx = u.v-\int v.du##
##=\ln x . \frac{1}{3}x^3-\int\frac{1}{3}x^3(\frac{1}{x}dx)##
##=\ln x . \frac{1}{3}x^3-\frac{1}{3}\int\frac{x^3}{x}dx##
##=\ln x . \frac{1}{3}x^3-\frac{1}{3}\int x^2dx##
##=\ln x . \frac{1}{3}x^3-\frac{1}{3}[\frac{1}{3}x^3+c]##
##=\ln x . \frac{1}{3}x^3-\frac{1}{9}x^3+c##
CMIIW.

The remaining two problems do not require integration by parts. They can be done by direct integration after some algebraic manipulation of the integrand.

How to do the algebraic manipulation? I don't understand. Give some hints again for the SECOND and THIRD problem.
 
  • #8
ehild
Homework Helper
15,543
1,913
expand the square in the second problem.
 
  • #9
95
0
expand the square in the second problem.

You're right!

##\int\frac{(x+1)^2}{x}dx##
##=\int\frac{x^2+2x+1}{x}dx##
##=\int\frac{x^2}{x}dx+\int\frac{2x}{x}dx+\int\frac{1}{x}dx##
##=\int x \ dx+2\int dx+\int\frac{1}{x}dx##
##=\frac{1}{2}x^2+2x+\ln x+c##

Is it correct?

One more problem to solve.

Give me a hint.
 
  • #10
ehild
Homework Helper
15,543
1,913
Write the numerator as the sum (x+3) -5.

You have the integral
##
\int \frac{(x+3)-5}{x+3}dx=\int 1-\frac{5}{x+3}dx
##
 

Related Threads on How To Integrate These?

  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
14
Views
1K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
425
  • Last Post
Replies
6
Views
677
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
4
Views
948
  • Last Post
Replies
19
Views
654
Top