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Results tagged with integration
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user 512834
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
1
vote
Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$
Note section:
\begin{align*}
\begin{gathered}
\therefore \sum_{n=0}^{\infty} \frac{(-1)^n H_n^{(2)}}{n^2}=-4 L i_4\left(\frac{1}{2}\right)+\frac{51}{16} \zeta(4)-\frac{7}{2} \ln (2) \zeta(3)+\ln ^2(2) …
- 224
1
vote
0
answers
101
views
Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$
I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x) …
- 224
5
votes
3
answers
818
views
Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$
I uploaded this question here and here from my old account.
Let $\psi^{(1)}$ be the trigamma function defined by
\begin{equation}
\tag{1}
\psi_1(z) = -\int\limits_0^1 \frac{x^{z-1} \ln x}{1-x} \, dx.
…
- 224
6
votes
1
answer
395
views
Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\l...
.$$
Approach:
Utilize the substitution $t=x/4$, integration by parts, and the identity $\sum_{n=1}^{\infty}H_{n}^{(2)}x^{n}=\frac{Li_{2}(x)}{1-x}$. …
- 224