Let $X_1, \ldots , X_n$ i.i.d with $X_1\sim \text{Exp}(5\theta)$. For the statistical distribution of $\theta$ the a priori belief is described by $\text{Gamma}(\alpha,1)$, where $a$ a positive parameter.
Find the Marginal probability density function for each $X_j$.
Find the estimator for $\theta$ according to Bayes considering the square loss function.
For the first question we have to calculate the integral $$f_{X_j}(x_j)=\int_0^{\infty}f_{x_1\ldots x_n}(x_1, \ldots , x_n)\, dx_1, \ldots dx_{j-1} dx_{j+1}\ldots dx_n$$ or not?
So first we have to find the function $f_{x_1\ldots x_n}(x_1, \ldots , x_n)$.
Knowing that $X_1\sim \text{Exp}(5\theta)$ does the following hold? $$f_{X_1\mid \Theta }(x\mid \theta)=\begin{cases}5\theta e^{-5\theta x_1} & \text{ for } x> 0 \\ 0 & \text{ for } x\leq 0\end{cases}$$
The MSE community suggests me to avoid long conversation in comments.
For the fisrt question below is a integral using gamma function, $$f_X(x)=\int_0^{+\infty}\prod_{i=1}^{n}(5\theta e^{-5\theta x_i}) \frac{1} {\Gamma(\alpha)}\theta^{\alpha-1}e^{-\theta}d\theta=\frac{5^n}{\Gamma{(\alpha)}}(1+5\sum_{i=1}^{n}x_i)\Gamma(n+\alpha)$$ Foe the second, Bayes estimation will be $$\theta=\mathbb{E}[\theta|X=x]=\frac{\int\theta f_\Theta(\theta)f_{X|\Theta}(x|\theta)d\theta}{\int f_\Theta(\theta)f_{X|\Theta}(x|\theta)d\theta}=(1+5\sum_{i=1}^{n}x_i)(n+1+\alpha)$$
For the UMVUE，a common method is to use Lehmann–Scheffé theorem, to check the distribution family is complete and from the fisrt question $S=\sum_{i=1}^{n}X_i$ is a complete statistic. And find a statistic $\Theta$ with $\theta$ expectation and then calculate $\mathbb{E}(\Theta|X)$