Please help me quickly

[yooshi]

Let ▁x = ( x_1 ,…..,x_(n ) ) be a random sample with size n taken from population has the distribution F(x,θ) = e^(-((x-β)/θ))/θ , β<x< ∞ find the moment estimator of θ , is it an unbiased estimator of θ ?!

I tried to resolve the matter but stopped at the integration
Will show you where i stopped and please help in completing the solution
Moment method :
(μ_r ) ̀=(M_r ) ̀
At r = 1 → (μ_1 ) ̀=(M_1 ) ̀
→ E(x) = ¯x
→ E(x) = ∫_β^∞〖x f(x,θ)dx〗 = ∫_β^∞〖x e^(-((x-β)/θ))/θ dx〗 =1/θ ∫_β^∞〖x e^(-((x-β)/θ)) dx〗 = ?????!!! here stopped
what i can do

[BGM]

This is a shifted exponential distribution. Let may let you to see this integral clearer as a Gamma integral.

Actually it just has the same distribution as where has an exponential distribution with mean equals to 1.