问题：计算不定积分

\[\int{\frac{x^n}{1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}}\text{d}x} \]

---

过程如下：记\(\displaystyle S\left( n \right) =1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}\)，对\(\displaystyle x\)求导，有\(\displaystyle S'\left( n \right) =S\left( n-1 \right)\)

因此有

\[\begin{align*} \int{\frac{x^n}{1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}}\text{d}x}&=\int{\frac{n!\left[ S\left( n \right) -S\left( n-1 \right) \right]}{S\left( n \right)}\text{d}x} \\ &=n!x-n!\int{\frac{S\left( n-1 \right)}{S\left( n \right)}\text{d}x} \\ &=n!x-n!\int{\frac{1}{S\left( n \right)}\text{d}S\left( n \right)} \\ &=n!x-n!\ln \left| 1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!} \right|+C \end{align*} \]