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MIT Integration Question:
Find: $$\int_{0}^{4} \begin{pmatrix} x \\ 5\end{pmatrix} dx$$ I know this might be somewhat simplified using the Gamma Function, however is there any simpler way to find this integral?

Math Attack
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DrVendetta
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  • ${n \choose 5}$ is just a constant. The answer is 4 times that. – Ninad Munshi Nov 24 '23 at 15:37
  • Do you mean $\int_0^4 \binom{x}{5} \ dx$? – HallaSurvivor Nov 24 '23 at 15:38
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    If you mean $\int_0^4\binom x5dx$ is just a polynomial. A bit annoying to compute with pen and paper, but rather straight forward. – ajotatxe Nov 24 '23 at 15:40
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    More specifically, for complex $n$ and natural $r$ we have the generalized definition (which is the only interpretation that makes sense here) that $\binom{n}{r} = \dfrac{n^{\underline{r}}}{r!}$ where $n^{\underline{r}}$ is the falling factorial notation. That is to say, $\binom{x}{5} = \dfrac{1}{5!}\times x(x-1)(x-2)(x-3)(x-4)$. Gamma function has nothing to do with this. – JMoravitz Nov 24 '23 at 15:42
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    Hint: King's rule implies this integral is $0$. – J.G. Nov 24 '23 at 15:42

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Just use the definition of the generalized binomial coefficient: $$\int_{0}^{4}\binom{x}{5}dx =\frac{1}{5!}\int_{0}^{4}x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)dx=0$$

JMoravitz
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Math Attack
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    To be clear, the definition does not in fact involve factorials with $x$, noting that $x$ might not be a whole number. – JMoravitz Nov 24 '23 at 15:43
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    @JMoravitz I know, but in fact the question is not even asked correctly given that the binomial coefficient is not a continuous function if only integers are to be considered, so the integral does not make sense. I think it's more of one of those "try having some fun by thinking outside the box" questions. – Math Attack Nov 24 '23 at 15:46
  • There does exist the generalized binomial coefficient which I alluded to already above in my comments which allows for any complex value for the top number, in particular any reals in the interval $[0,4]$. The end result is that your final expression you write is the correct simplification, but you got there by using an incorrect definition and interpretation of the binomial coefficient. – JMoravitz Nov 24 '23 at 15:47
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    Again, that is wrong. The generalized binomial coefficient $\binom{x}{r}$ is defined for any complex $x$ and natural $r$ and when treated as a function of $x$ is indeed continuous. – JMoravitz Nov 24 '23 at 15:51