evaluation of this logarithmic integrals

Question

what is the value $$ \int_{a}^{\infty} \frac{\log^{n}(x)}{x^{2}}\mathrm{d}x $$

'a' is a positive integer and so is 'n'

my gues with a change of variable $ x=e^{t} $ is that this integral would be related to the incomplete gamma function $$ \Gamma (n.\log(a)) $$

@Julián Aguirre. I am living next to you ! Un abrazo. – Claude Leibovici Nov 20 '14 at 16:15 — Claude Leibovici, Nov 20 '14 at 16:15
I am living next to you too ! Un abrazo. – Claude Leibovici Nov 20 '14 at 16:36 — Claude Leibovici, Nov 20 '14 at 16:36

score 2 · Answer 1 · answered Nov 20 '14 at 16:13

2

Hint

Integrate by parts using $u=\log^n(x)$, $dv=\frac{dx}{x^2}$. You will arrive to a splendid recurrence relation.

I am sure that you can take from here.

answered Nov 20 '14 at 16:13

Claude Leibovici

score 0 · Accepted Answer · answered Nov 20 '14 at 16:10

0

Mathematica gave:

$$\int_{a}^{\infty} \frac{\log^{n}(x)}{x^{2}}\mathrm{d}x=\Gamma(n+1,\log(a))$$

answered Nov 20 '14 at 16:10

mike

2 Answers2