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Question: Integrate$$\int\limits_0^1dx\,\frac {\log(1-x)\log^2(1+x)}x=-\frac {\pi^4}{240}$$

I'm curious as to if there is a way to integrate this. I've tried using integration by parts to get$$I=-\frac {\pi^2}6\log^22+2\int\limits_0^1dx\,\frac {\operatorname{Li}_2(x)\log(1+x)}{1+x}$$However, I'm not sure how to continue even further. The polylog in the second integrand seems a bit intimidating and I don't see how the first term even helps.

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The integral is hard to tackle directly (without using Euler sums), but there is a nice trick (which is literally the same as posed above).


Let $$I = \int_0^1 {\frac{{\ln (1 - x){{\ln }^2}(1 + x)}}{x}dx} \qquad J = \int_0^1 {\frac{{{{\ln }^2}(1 - x)\ln (1 + x)}}{x}dx} $$

We have $$\begin{aligned} 3I + 3J + \int_0^1 {\frac{{{{\ln }^3}(1 - x)}}{x}dx} + \int_0^1 {\frac{{{{\ln }^3}(1 + x)}}{x}dx} &= \int_0^1 {\frac{{{{\ln }^3}(1 - {x^2})}}{x}dx} \\ &= \frac{1}{2}\int_0^1 {\frac{{{{\ln }^3}(1 - u)}}{u}du} \end{aligned}$$ the substitution $x^2 = u$ is used. Hence $$\tag{1}3I + 3J + \int_0^1 {\frac{{{{\ln }^3}(1 + x)}}{x}dx} = \frac{{{\pi ^4}}}{{30}}$$

On the other hand, $$\begin{aligned}\int_0^1 {\frac{{{{\ln }^3}(1 - x)}}{x}dx} - 3J + 3I - \int_0^1 {\frac{{{{\ln }^3}(1 + x)}}{x}dx} &= \int_0^1 {\frac{{{{\ln }^3}(\frac{{1 - x}}{{1 + x}})}}{x}dx} \\ &= \int_0^1 {\frac{{2{{\ln }^3}u}}{{(1 - u)(1 + u)}}du} \\ &= \int_0^1 {\frac{{{{\ln }^3}u}}{{1 - u}}du} + \int_0^1 {\frac{{{{\ln }^3}u}}{{1 + u}}du} \end{aligned}$$ the substitution $u=\frac{1-x}{1+x}$ is used. Giving $$\tag{2} - 3J + 3I - \int_0^1 {\frac{{{{\ln }^3}(1 + x)}}{x}dx} = - \frac{{7{\pi ^4}}}{{120}} $$

Adding $(1)$ and $(2)$ together gives $I=-\frac{\pi^4}{240}$.

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  • $\begingroup$ Sir, could you explain me how did you get π^4/30 in (1) and -7π^4/120 in (2)? $\endgroup$ – William May 23 '18 at 9:09
  • $\begingroup$ @William Use the values $$\int_0^1 \frac{\ln^3 u}{1-u} du = -\frac{\pi^4}{15} \qquad \int_0^1 \frac{\ln^3 u}{1+u} du = -\frac{7\pi^4}{120}$$ which follows easily from Riemann zeta function. $\endgroup$ – pisco May 23 '18 at 16:57

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