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.

Question

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?

Answer 1

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!

Answer 2

Hint: If $p_n(x)=1+x+\cdots+x^n/n!$ then $p'_n(x)=p_n(x)-x^n/n!$. Therefore $$ \frac{x^n}{p_n(x)}=n!\frac{p_n(x)-p'_n(x)}{p_n(x)}. $$