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· **Challenge System:** A system in which you challenge monsters one by one.

· **Multiplayer:** The first multiplayer fantasy role playing game being released in Japan.

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· **Icons:** A system in which you refer to your items and monsters in an easy-to-understand manner with icons that appear on the screen.

· **Eternal Happiness:** In addition to powering up your character, you can also choose to invest your magic into various items to consume them.

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Q:

Integral : $\int_0^1\frac{(1-x)^{n-1}}{(n-1)!}dx$

In the following expression :
$$\int_0^1\frac{(1-x)^{n-1}}{(n-1)!}dx$$
This can be easily integrated and the result is : $$\frac{(-1)^n}{n!}$$
How can the following be done?
$$\int_0^1\frac{(1-x)^{n-1}}{(n-1)!}dx$$
Thank you for your help.

A:

\begin{align}
\int_0^1 \frac{(1-x)^{n-1}}{(n-1)!} \; \mathrm{d}x
&= \int_0^1 \sum_{k=0}^{n-1} \binom{n-1}{k} (-1)^k x^{n-1-k} \; \mathrm{d}x \\
&= \sum_{k=0}^{n-1} \binom{n-1}{k} (-1)^k \int_0^1 x^{n-1-k} \;

 

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