"Continuous" binomial distribution

Hi.

I want to use some kind of "continuous" binomial distribution. I know binomial distribution is defined for natural numbers, but I want to extend it to the use of real numbers.

I also know that the gamma function is an extension of factorial to be used with real numbers.

My question is:

Is a probability density function? More precisely, is the integral from 0 to n equal to 1?

Playing with excel it doesn't seem to be 1, but i'm not sure. Do you know what function should i use?

Thanks.