Need help on a convergent divergent problem! (int, 1-infty, (X^p)ln(x)dx)

[DestoryJar]

Hey guys,

I am currently working on a problem where I can not integrate due to an extra variable; I'm sure I'm most likely making a mistake, but I just can't seem to figure it out.

Question:

For what values of p is the following integral convergent?

Integral of (X^p)ln(x)dx from 1 to infinite.

My work so far (int is just shorthand for integral):

u=ln(x) du=(1/x)dx dv=x^p v=(x^(p+1))/(P+1) --> ln(x)x^p - int((x^(p+1))/((P+1)(1/x)dx))

From this point, I do not know how to solve using a partial fraction decomposition due to p making it impossible to solve for A, B, C, etc..

If anyone can help me or give me an alternate means of finding a solution, that would be great!!!

Thanks,
DestoryJar

 

[HallsofIvy]

What you have done so far is very good. One obvious point is that you do not need a "partial fraction decomposition" because there is no denominator! \(\displaystyle \frac{x^{p+1}}{x}= x^p\). Your last integral is just \(\displaystyle \frac{1}{p+1}\int_1^\infty x^p dx\).

(The way you have the parentheses on the last integral is wrong- you have dx in the denominator!)

 

[DestoryJar]

What you have done so far is very good. One obvious point is that you do not need a "partial fraction decomposition" because there is no denominator! \(\displaystyle \frac{x^{p+1}}{x}= x^p\). Your last integral is just \(\displaystyle \frac{1}{p+1}\int_1^\infty x^p dx\).

(The way you have the parentheses on the last integral is wrong- you have dx in the denominator!)

Thank you so much!

I was looking at p like it was a variable instead of a constant!