Find marginal distribution of Y while knowing distribution of X and $Y|X$
Your integral is correct. One can show that $$\int_0^1 x^y (1-x)^{n-y} \, dx = \frac{y! (n-y)!}{(n+1)!},$$ see the Beta function or order statistics of i.i.d. uniform random variables. Multiplying by $\binom{n}{y} = \frac{n!}{y! (n-y)!}$ and canceling gives a simple expression for $P(Y=y)$.