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Next: Factorials and the Gamma Up: Finding Previous: Extending the result

Final result

We have proved that

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

and the more general result

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

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


Alexander Frolkin 2001-06-02