The background of the problem is as follows:
Suppose $X_1,...,X_n \sim U(a,b)$ independently where $a$ and $b$ are unknown parameters and $a < b$. Let $\hat\tau$ be the MLE of $\tau$, where $\tau = \int x dF(x)$.
How to analytically find the standard error of $\hat\tau$?
As mentioned in comments, $\tau=E(X_1)=\frac{a+b}{2}$.
If $(\hat a,\hat b)$ is the MLE of $(a,b)$, then by invariance property MLE of $\tau$ is $$\hat\tau=\frac{\hat a+\hat b}{2}$$
Verify that $\hat a=X_{(1)}$ and $\hat b=X_{(n)}$, the minimum and maximum of the sample observations respectively.
You are looking for the standard error (standard deviation) of the statistic $\hat\tau$. So equivalently find the variance of $\hat\tau$ and take the positive square root.
$$\operatorname{Var}(\hat\tau)=\frac{1}{4}\left[\operatorname{Var} X_{(1)}+\operatorname{Var}X_{(n)}\right]+\frac{1}{2}\operatorname{Cov}(X_{(1)},X_{(n)})$$
The distributions of $X_{(1)}$ and $X_{(n)}$ are straightforward to derive, from which you can find the mean and variance of each. For the covariance, refer to the joint distribution (also easy to derive) of $(X_{(1)},X_{(n)})$:
$$g(x,y)=n(n-1)(F(y)-F(x))^{n-2}f(x)f(y)\,\mathbf1_{x<y}$$
, where $F$ and $f$ is the population cdf and pdf respectively.
In a different approach, you can derive the sampling distribution of $\hat\tau$ (sometimes called the sample mid-range) directly and find the standard error. Here is a relevant thread.
You might find it easier to define $Y_i=\frac{X_i-a}{b-a}\sim U(0,1)$ and then work with the $Y_i$'s in both methods.
