计算积分

河中巨怪

\[\int_{0}^{1}{\frac{{{\ln }^{n }}x}{{{x}^{2}}+x+1}}\,\mathrm{d} x, \,\text{for}\, \,n \in\mathbb{N^*}.\]

\[\int_{0}^{1}{\frac{{{\ln }^{n }}x}{{{x}^{2}}-x+1}}\,\mathrm{d} x, \,\text{for}\, \,n \in\mathbb{N^*}.\]

Renascence_5

for the first one,give you a hint. Using $$\frac{1}{x^2+x+1}=\frac{1-x}{1-x^3}$$ and then the geometric series.maybe you will get a series related to the digamma function or polygarithm.

青青子衿

某群一道积分:计算
\begin{align*}
\int_{0}^{1}{\frac{\ln\,\!x}{{{x}^{2}}+x+1}}\mathrm{d}x&=\dfrac{4\pi^3}{81\sqrt3}\\
\int_{0}^{1}{\frac{\ln\,\!x}{{{x}^{2}}-x-1}}\mathrm{d}x&=\dfrac{\pi^2}{5\sqrt5}\\
\end{align*}

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2020-9-21 23:07