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I've stumbled onto this general integral that has closed form values for the $n\in \Bbb{Z^+}$ $$I_n=\int_0^\infty \frac{\ln^n(x+1)-\ln^n(x)}{x+1}dx$$

Obviously $I_0=0$ but higher values of $n$ yield interesting results. The first few are: $$I_1=\frac{\pi^2}{6}$$ $$I_2=0$$ $$I_3=\frac{7\pi^4}{60}$$ $$I_4=0$$ $$...$$ So it appears that $I_{2n}=0$ and that $I_{2n+1}=C_{2n+1}\zeta\left(2(n+1)\right)$ where $C_{n}$ is some rational number that is equal to $0$ when $n\in2\Bbb{Z^+}$. Can anyone explain why $I_{2n}=0$ and how one can derive $I_{2n+1}$?

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  • $\begingroup$ Nice problem! +1 I find $I_n = \left(2-4^{-n}\right) (2 n+1)! \zeta ((2 n+1)+1)$ $\endgroup$ Oct 17, 2023 at 10:11

2 Answers 2

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$I_n=\int_0^\infty\frac {\ln^n(x+1)-ln^n(x)}{x+1}dx=\int_1^\infty\frac{\ln^n(x)}{x(x+1)}dx-\int_0^1\frac{\ln^n(x)}{1+x}dx$, using $x+1 \to x$ in the $ln^n(x+1)$ term.

Let $y=\frac{1}{x}$ in the first integral and it becomes $\int_0^1\frac{(-\ln(y))^n}{1+y}dy$.

Net result $I_n=0$ for even $n$ and $I_n=-2\int_0^1\frac{\ln^n(x)}{1+x}dx$ for odd $n$. This last integral can be found in Gradshteyn and Ryzhik Table of Integrals...

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  • $\begingroup$ Can you list what the value for the integral is, because I cannot find it in any tables. Or can you link to where the value is stated? $\endgroup$
    – aleden
    May 2, 2019 at 1:15
  • $\begingroup$ Nevermind, Wolfram can evaluate it and the values match up. Thank you for your answer! $\endgroup$
    – aleden
    May 2, 2019 at 3:59
  • $\begingroup$ From the Table I referenced: $\int_0^1\frac{(lnx)^{2n-1}}{1+x}dx=\frac{1-2^{2n-1}}{2n}\pi^{2n}|B_{2n}|$ where $B_{2n}$ are Bernoulli numbers. If you can access that table, look for section 4.271. $ $\endgroup$ May 2, 2019 at 19:38
  • $\begingroup$ I see, I also realized that the integral you had $I_n$ reduced to can be evaluated by a simple substitution to yield the answer in terms of the Gamma and Dirichlet Eta Function. $\endgroup$
    – aleden
    May 2, 2019 at 19:40
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I have found a closed form for the integral in question

$$I(n) = \int_0^\infty \frac{\ln^n(x+1)-\ln^n(x)}{x+1}\,dx\tag{1}$$

with a method which quite is common but has not been used in the partial solutions provided here by others.

We start with the generating integral

$$g(a) = \int_0^{\infty } \frac{(x+1)^a-x^a}{x+1} \, dx\tag{2}$$

from which the $I(n)$ can be generated by differentiating $n$ times with respect to $a$ at $a=0$

$g(a)$ can be calculated explicitly: letting $x\to \frac{t}{1-t}$ gives

$$\int_0^1 (1-t)^{-a-1} \left(1-t^a\right) \, dt = \pi \csc (\pi a)-\frac{1}{a} \tag{3}$$

But here we have a nice series expansion

$$\pi \csc (\pi a)-\frac{1}{a}=-2 a \sum _{k=1}^{\infty } \frac{(-1)^k}{k^2-a^2}\tag{4}$$

Expanding the summand into a geometric series

$$-2a \sum _{k=1}^{\infty } \frac{ (-1)^k}{k^2-a^2}=-2a\sum _{k=1}^{\infty } \frac{(-1)^k}{k^2 \left(1-\left(\frac{a}{k}\right)^2\right)}\\ = -2 a\sum _{k=1}^{\infty } \sum _{n=0}^{\infty }\frac{(-1)^k }{k^2}\left(\frac{a}{k}\right)^{2 n}=-2 \sum _{n=0}^{\infty } a^{2 n+1} \sum _{k=1}^{\infty } \frac{(-1)^k}{k^{2 n+2}}$$

and doing the $k$-sum gives this power series in $a$

$$g(a) = \sum _{n=0}^{\infty } \left(2-4^{- n}\right) a^{2 n+1} \zeta (2 n+2))\tag{5}$$

from which we find this closed expression for our integral:

$$I(2n) = \left(\frac{\partial}{\partial a}\right)^{2n} g(a)|_{a\to 0}=0\tag{6a}$$ $$I(2n+1) =\left(\frac{\partial}{\partial a}\right)^{2n+1} g(a)|_{a\to 0}= \left(2-4^{-n}\right) (2 n+1)! \zeta (2 n+2)\tag{6b}$$

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