Mathematics - Indefinite Integration Question with Solution | TestHub

Let $g(x)=1+x+\dots +\frac{x^n}{n!}$

$$\begin{array}{rl}{g}^{\prime }(x)& =1+x+\dots +\frac{x^{n-1}}{(n-1)!}\\ \Rightarrow x^n& =n!(g(x)-g^{\prime }(x))\\ \Rightarrow I& =\int \frac{x^n}{1+x+\dots +\frac{x^n}{n!}}dx\\ & =n!\int \frac{g(x)-g^{\prime }(x)}{g(x)}dx\\ & =n!\left[x-{\log }_e(g(x))\right]+C\\ f(x)& =\ln \left(1+x+\frac{x^2}{2!}+\dots +\frac{x^n}{n!}\right)\end{array}$$