Q&A

# What is a dummy variable (in an integral)?

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While I was learning about Gamma and Beta functions, I saw that they wrote that $x$ is dummy variable. What did they mean by that? I know that $x$ is changeable.

$$\Gamma[n]=\int_0^\infty e^{-x}x^{n-1} \mathrm dx$$

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When integrating, a dummy variable is one that "disappears completely in the final result."

For example, in the expression $\int x\ \mathrm{d}x$, $x$ is not a dummy variable because the expression is equivalent to $\frac{x^2}{2}+c$, that is, the $x$ hasn't "disappeared." If we had replaced $x$ with, say, $y$, then the result would change (to $\frac{y^2}{2}+c$).

But in the expression you wrote, $x$ is a dummy variable because the expression is equivalent to $\Gamma[n]$, and the $x$ has "disappeared." If we had replaced $x$ with, say, $y$, then the result would not change (it would still be $\Gamma[n]$).

(Note that statistics has a different meaning for the term dummy variable.)

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