# Questions tagged [beta-function]

For questions about the Beta function (also known as Euler's integral of the first kind), which is important in calculus and analysis due to its close connection to the Gamma function. It is advisable to also use the [special-functions] tag in conjunction with this tag.

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### I want to know the expectation of a product of independent beta distributed random variables.

I have an equation of the form $$Z = \frac{\prod_{i=1}^p X_i}{\prod_{i=1}^p Y_i}$$ where $$X_i \sim \mathcal{B}(\alpha_{x_i},\beta_{x_i})$$ and $$Y_i \sim \mathcal{B}(\alpha_{y_i},\beta_{y_i})$$ $X_i$ ...
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### Evaluate the indefinite integral $\int\left(1-\left(1-x^{\frac{1}{a}}\right)^a\right)\mathrm dx\,$

I am currently working on the following expression: $$\int\left(1-\left(1-x^{\frac{1}{a}}\right)^a\right)\mathrm dx$$ I am particularly interested in finding the closed-form solution for this integral....
### Proving that $I=\int_0^\infty \frac{n x^{n-1}}{(1+x)^{n+1}}dx<+\infty$
I got this integral from a probability question in which I am interested to prove that it is $<\infty$ $$I=\int_0^\infty \frac{n x^{n-1}}{(1+x)^{n+1}}dx,~~~n \ge 1$$ I tried substitution $y =x+1$, ...