# Integration involving Gamma Distribution

1. Oct 27, 2009

### cse63146

1. The problem statement, all variables and given/known data
Evaluate $$\int^{\infty}_0 \frac{1}{B^{\alpha} \Gamma(\alpha)} \theta^{\alpha + x -1}e^{- \theta(1+B)/B} d \theta$$

2. Relevant equations

3. The attempt at a solution

$$\int^{\infty}_0 \frac{1}{B^{\alpha} \Gamma(\alpha)} \theta^{\alpha + x -1}e^{- \theta(1+B)/B}d \theta$$

$$= \frac{1}{B^{\alpha} \Gamma(\alpha)} \int^{\infty}_0 \theta^{\alpha + x -1}e^{- \theta(1+B)/B}d \theta$$

Let $$u = \theta(1+B)/B$$ and $$du = \frac{1 + B}{B} d\theta$$

$$= \frac{1}{B^{\alpha} \Gamma(\alpha)} \int^{\infty}_0 \frac{1 + B}{B} \frac{Bu}{1+B}^{\alpha + x -1}e^{- u} du$$

I'm stuck here. Not sure what to do next.