Calculate $\int \limits {x^n \over 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}} dx$ where $n$ is a positive integer. Calculate $$\int \limits {x^n \over 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}} dx$$ where $n$ is a positive integer.
Would you give me a hint?

Ooh, the solution is actually cute! Let's write $P_n$ for the denominator. The numerator $x^n$ is a simple linear combination of $P_n$ and $P_n'$:
$$ x^n = n! (P_n - P_n').$$
Using this, you can simplify to a form which is very easy to integrate!
