# ∫x*lnx*dx homework help

1. Mar 31, 2009

### brandy

1. The problem statement, all variables and given/known data
find the derivative of x*lnx
and hence find
∫x*lnx*dx

3. The attempt at a solution
the derivative is 1+lnx
the integral (by parts) i worked out to be x^2 * ln(x-1/2) / 2 + c

but i dont know how you find the integral from the derivative.
help?
also is my integration by parts correct?

Last edited: Mar 31, 2009
2. Mar 31, 2009

### CompuChip

Re: ∫x*lnx*dx

"find the derivative of x*lnx*dx"
What is x*ln(x)*dx?

∫x*lnx*dx is a function, whose derivative is x*ln(x).
x*lnx is a function, whose derivative is 1 + ln(x).

As for your integration by parts, how on earth did you manage to change ln(x) to ln(x - 1/2) ?

3. Mar 31, 2009

### brandy

Re: ∫x*lnx*dx

both were typos

there should be no .dx for the derivative
and the integral should be x^2(lnx - 1/2) / 2 + c

oops

sorry

now will anyone answer my question?

4. Mar 31, 2009

### D H

Staff Emeritus
Re: ∫x*lnx*dx

The question is asking you to use the fact that $\frac{d}{dx}(x \ln x) = 1+\ln x$ in calculating $\int x \ln x\,dx$.

How did you do your integration by parts?

5. Mar 31, 2009

### brandy

Re: ∫x*lnx*dx

how do you use that fact!!!!

i went u=lnx u'=1/x
v'=x v=x^2/2
∫x*lnx*dx=x^2lnx / 2 - ∫x^2/2x
x^2lnx / 2 - ∫x/2
x^2lnx / 2 - x^2/4
rearange
X^2 * (lnx - 1/2)*1/2

6. Mar 31, 2009

### D H

Staff Emeritus
Re: ∫x*lnx*dx

If I told you that I would be telling you how to solve the problem.

Knowing f(x) and its derivative f'(x), how would you compute ∫ f(x) dx via integration by parts?

7. Mar 31, 2009

### brandy

Re: ∫x*lnx*dx

but for parts u only need the derivative of lnx not x*lnx

8. Mar 31, 2009

### D H

Staff Emeritus
Re: ∫x*lnx*dx

You are supposed to use the derivative of x*ln(x) to aid in the calculation of ∫ x*ln(x) dx. Yes, you can calculate this integral by other means. However, in doing so you will not be doing what was asked of you and you will get minimal points as a result.

The rules of this site (very good rules) prohibit me from giving you the solution. I can however ask some very leading questions. You have not answered my question from post #6:

Knowing f(x) and its derivative f'(x), how would you compute ∫ f(x) dx via integration by parts?

9. Apr 1, 2009

### tiny-tim

Hi brandy!
Well, you obviously need to put (1 + lnx) in there somewhere, before you start.