Learn to Find Derivatives and Integrals of x*lnx | Homework Help

[brandy]

Homework Statement

find the derivative of x*lnx
and hence find
∫x*lnx*dx

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 don't know how you find the integral from the derivative.
help?
also is my integration by parts correct?

please and thankyou

 

[CompuChip]

"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) ?

 

[brandy]

both were typos
my bad

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?

 

[D H]

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?

 

[brandy]

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

 

[D H]

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?

 

[brandy]

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

 

[D H]

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?

 

[tiny-tim]

Hi brandy!

how do you use that fact!
…
∫x*lnx*dx = …

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