Caculating integral with binomial coefficient

cielo
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Homework Statement





Homework Equations


What is the integral of

\int^{0}_{1} nCy x^{y} (1-x)^{n-y} dx ?


The Attempt at a Solution


\left(nCy\right) \int^{0}_{1} x^{y} (1-x)^{n-y} dx
 
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It looks like a beta function to me.
 


Dick said:
It looks like a beta function to me.

Thank you for your good idea Dick! =)
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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