I can find $\int{\binom{x}{1}}dx$,$\int{\binom{x}{2}}dx$, etc. Using the definition that I know as $$\binom{n}{k} = \frac{n!}{(n-k)!k!}$$ I am trying to find $$\int{\binom{x}{n}}dx$$ in terms of both x and n, where n is just a number. So far I have found that $$\int{\binom{x}{n}}dx = \frac{1}{n!}\int\prod_{k=0}^{n-1}(x-k)dx$$ but I have no idea where to go from here. I’ve heard that the product within the integral is called a Pochhammer symbol, but I’m not sure how it can be integrated. How can I integrate this - or is it not possible to find a formula in terms of x and n (plugging $n=1$ to the formula should give $\frac{x^2}{2} + C$, $n=2$ should give $\frac{x^3}{6} - \frac{x^2}{4}+C$, etc.)?


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