Find $E(X\mid Y)$ if $Y$ is beta and $X$ conditionally on $Y$ is geometric with parameter $Y$

Question

Let be $Y$ Beta$(\alpha, \beta )$ distributed random variable. Further more let $X$ conditional on $\{Y=y \}$ geometric distributed with paramter $y$. I want to determine $\mathbb E( X\mid \sigma(Y))$.

My attempt, $$\int_{0}^{1}x f(x\mid y)\,dx= \int_{0}^{1} x(1-(1-y)^xy\, dx =\int_{0}^{1}x \, dx - \int_{0}^{1}(1-y)^{x+1}y\, dx$$ and we now $\int^{1}_0(1-y)^{x+1}y$ is beta distributed.Does this approach make sense? Help is much appreciated. Especially I am not sure about the borders aswell.