Computing the integral $\int_1^n\frac{1}{(\ln x)^p}dx$

Question

What is the approach I should have? How can I calculate this integral? I put it into Mathematica in order to get a step by step solution, but Mathematica does not give a step by step explanation. The Wolfram solution incorporates a Gamma function but I don't know it.

Do I need to know this Gamma function to proceed?

$$\int_1^n\frac{1}{(\ln x)^p}dx$$

Thank you.

I tried substitution and then integration by parts... But I could not solve it... — Charlie, Jan 24 '14 at 12:02

Norbert · Answer 1 · 2014-01-24T12:09:29.147

1

You do need Gamma function after substitution $t=\ln x$ and even more. Mathematica gives a reslt but in trems of special functions. $$ \log ^{-p}(n) \left(p \Gamma (-p) (-\log (n))^p-\log (n) E_p(-\log (n))\right) $$ Here $E_p$ is an exponential integral function.

edited Jan 24 '14 at 12:09

answered Jan 24 '14 at 12:04

Norbert

56,803

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

1

I think the issue here is that you need special functions to be able to express this integral in closed form, and without them you cannot. The reason for this is that some integrals cannot be expressed solely in terms of elementary functions.

edited Apr 13 '17 at 12:21

Community

1

answered Jan 24 '14 at 12:18

preferred_anon

17,121

Computing the integral $\int_1^n\frac{1}{(\ln x)^p}dx$

2 Answers2