计算

In=∫01xmln⁡nxdx\large I_n=\int_0^1{x^m\ln ^nx}dx

设 InI_n 的指数生成函数为 G(t)G(t), 则

G(t)=∑n=0∞Inn!tn{\large G\left( t \right) =\sum_{n=0}^{\infty}{\frac{I_n}{n!}}t^n }\\

计算过程如下

G(t)=∑n=0∞Inn!tn=∑n=0∞∫01xmln⁡nxdxn!tn=∫01xm∑n=0∞ln⁡nxn!tndx=∫01xm∑n=0∞(tln⁡x)nn!dx=∫01xmetln⁡xdx=∫01xmxtdx=∫01xm+tdx=1m+t+1=1m+111+tm+1=1m+1∑n=0∞(−1)n⋅n!n!(tm+1)n=∑n=0∞(−1)n⋅n!n!1(m+1)n+1⋅tn=∑n=0∞1n!⋅(−1)n⋅n!(m+1)n+1⋅tn\large \begin{aligned} G\left( t \right) &=\sum_{n=0}^{\infty}{\frac{I_n}{n!}}t^n \\ &=\sum_{n=0}^{\infty}{\frac{\int_0^1{x^m\ln ^nx}dx}{n!}}t^n \\ &=\int_0^1{x^m\sum_{n=0}^{\infty}{\frac{\ln ^nx}{n!}}t^n}dx \\ &=\int_0^1{x^m\sum_{n=0}^{\infty}{\frac{\left( t\ln x \right) ^n}{n!}}}dx \\ &=\int_0^1{x^me^{t\ln x}}dx \\ &=\int_0^1{x^mx^t}dx \\ &=\int_0^1{x^{m+t}}dx \\ &=\frac{1}{m+t+1} \\ &=\frac{1}{m+1}\frac{1}{1+\frac{t}{m+1}} \\ &=\frac{1}{m+1}\sum_{n=0}^{\infty}{\frac{\left( -1 \right) ^n\cdot n!}{n!}\left( \frac{t}{m+1} \right) ^n} \\ &=\sum_{n=0}^{\infty}{\frac{\left( -1 \right) ^n\cdot n!}{n!}\frac{1}{\left( m+1 \right) ^{n+1}}\cdot t^n} \\ &=\sum_{n=0}^{\infty}{\frac{1}{n!}\cdot \frac{\left( -1 \right) ^n\cdot n!}{\left( m+1 \right) ^{n+1}}\cdot t^n} \end{aligned}

易得

In=(−1)n⋅n!(m+1)n+1.\large I_n=\frac{\left( -1 \right) ^n\cdot n!}{\left( m+1 \right) ^{n+1}}.\\