For $X_{i}\sim UNIF(\theta_{1},\theta_{2})$, we know that $X_{1:n}$ and $X_{n:n}$ are jointly sufficient for $\theta_{1}$ and $\theta_{2}$.
Suppose that it is desired to estimate the mean $\mu = \frac{\theta_{1}+\theta_{2}}{2}$, to do this, let's find a UMVUE for $\mu$:
To see that this UMVUE is $\frac{X_{1:n}+X_{n:n}}{2}$ , using the Lehmann-Scheffé Theorem, it would be sufficient to show that $\frac{X_{1:n}+X_{n:n}}{2}$ is unbiased for $\mu$.
My attempt:
Lets find $\mathbb{E}(X_{k:n})$ to calculate $\mathbb{E}(X_{1:n})$ and $\mathbb{E}(X_{n:n})$:
Pdf of $X_{k:n}$:
$$\frac{n!}{(k-1)!(n-k)!}f(x)[F(x)]^{k-1}[1-F(x)]^{n-k}$$
but $f$ corresponds to $UNIF(\theta_{1},\theta_{2})$, so:
$$\mathbb{E}(X_{k:n})=\frac{n!}{(k-1)!(n-k)!(\theta_{2}-\theta_{1})}\int_{\theta_{1}}^{\theta_{2}} \left[\frac{x-\theta_{1}}{\theta_{2}-\theta_{1}}\right]^{k-1}\left[1-\frac{x-\theta_{1}}{\theta_{2}-\theta_{1}}\right]^{n-k} \,dx$$
Let $u=\frac{x-\theta_{1}}{\theta_{2}-\theta_{1}}$, then:
$$\mathbb{E}(X_{k:n})=\frac{n!}{(k-1)!(n-k)!}\int_{0}^{1} u^{k-1}(1-u)^{n-k} \,du$$
How can I continue with the integral? I've tried using the gamma function without much success.