Order Statistics: Expectations and Densities for Uniform Samples

Let $X_1, X_2, \ldots, X_n$ be $n$ i.i.d. uniform random variables on $[0, 1]$. Denote their order statistics $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$. (a) Prove that $E[X_{(k)}] = \dfrac{k}{n+1}$ for any $k \in \{1, 2, \ldots, n\}$. (b) Derive the probability density function of $X_{(k)}$ directly -- without first finding the CDF -- and identify the resulting distribution by name and parameters.