Final result

We have proved that

$$\int_0^x t^n {\rm e}^t {\rm d}t = {\rm e}^x \sum_{r=0}^n (-1)^{n-r} \frac{n!}{r!} x^r - (-1)^n n!$$ (7)

and the more general result

$$\int_0^x t^n {\rm e}^{at} {\rm d}t = {\rm e}^{ax} \sum_{r=0}^n (-1)^{n-r} \frac{n!}{r!} \frac{x^r}{a^{n-r+1}} - \frac{(-1)^n n!}{a^{n+1}}$$ (8)

These agree with the first few $I_n(x)$ expressions found at the beginning.