Improper Integral of ∫dx/(x^a 〖(lnx)〗^b ) from 0 to ∞ - Homework Help

[elabed haidar]

Homework Statement

the integral of ∫dx/(x^a 〖(lnx)〗^b ) from zero to infinite

Homework Equations

The Attempt at a Solution

i divided the equation to two parts I and J
where I is from 0 to 1
and J from 1 to infinite then i tried to to neighbourhood of zero i couldn't know how
i tried to do comparision test , well you can't because even if a and b are postive you can't know which is bigger? i need a detailed answer please help and soon
thank you

 

[Ray Vickson]

Please re-enter your integral in plain AASCI: on my reader it has unprintable characters, so what I see is
\int dx/(x^a \mbox{something} (\ln x)\mbox{something}^b

RGV

 

[elabed haidar]

this integral is right without the something its only a function of x and lnx i need details in solving it

 

[Char. Limit]

The integral in question is:

\int_0^\infty \frac{dx}{x^a log^b(x)}

It's solvable using the incomplete gamma function.

 

[micromass]

Do you need the exact answer to your integral (in that case, do you know complex analysis?) or do you simply need to know that the integral converges?

 

[elabed haidar]

i need to know when does it converge please and if you guys don't mind with details , i tried the neighbourhood method , comparision test but I am getting my stelf into bigger problems

 

[micromass]

Get rid of that annoying logarithm first. Substitue x=e^t. What do you have left, then?

 

[elabed haidar]

thank you very much now the neighbourhood towards zero is much easier thanks