Integration: How to decide which formula to use? I want to calculate the integration of,
$$
\frac{1}{y! \Gamma(\alpha) \beta^\alpha}\int_{0}^{\infty} \lambda^{y+\alpha-1} \exp\left\{ \frac{-\lambda}{\frac{\beta}{\beta+1}} \right\} d\lambda \ \ \ \ \ \ \ \ \ \ (1)
$$
The answer says, $$ \begin{pmatrix}y + \alpha -1 \\ y \end{pmatrix}  \left( \frac{\beta}{\beta+1}\right)^y \left( \frac{1}{\beta+1} \right)^\alpha$$
I am not sure where to start with. When we encounter this type of integration, what is the first thing we should consider? And how can I calculate Equation (1)?
 A: By definition of Gamma Function,
$$\Gamma (n)=\int\limits_{0}^{\infty} t^{n-1} e^{-t}dt$$
$$
I=\frac{1}{y! \Gamma(\alpha) \beta^\alpha}\int_{0}^{\infty} \lambda^{y+\alpha-1} \exp\left\{ \frac{-\lambda}{\frac{\beta}{\beta+1}} \right\} d\lambda \ \ \ \ \ \ \ \ \ \ 
$$
Take subsitution $s=\frac{\beta+1}{\beta}\lambda$,
$
I=\frac{1}{y! \Gamma(\alpha) \beta^\alpha}\int_{0}^{\infty} (\frac{s \beta}{\beta+1})^{y+\alpha-1} \exp{(-s)} (\frac{\beta ds}{\beta+1}) 
$
$
=\frac{1}{y! \Gamma(\alpha) \beta^\alpha}\bigg(\frac{\beta}{\beta+1}\bigg)^{y+\alpha}\int_{0}^{\infty} s^{y+\alpha-1} \exp{(-s)} ds 
$
$
=\frac{1}{y! \Gamma(\alpha) \beta^\alpha}\bigg(\frac{\beta}{\beta+1}\bigg)^{y+\alpha}\Gamma(y+\alpha)
$
$
=\frac{1}{y! \Gamma(\alpha) \beta^\alpha}\bigg(\frac{\beta}{\beta+1}\bigg)^{y}\bigg(\frac{\beta}{\beta+1}\bigg)^{\alpha}\Gamma(y+\alpha)
$
$
=\frac{(y+\alpha -1)!}{y! (y+\alpha-1-y)!}\bigg(\frac{\beta}{\beta+1}\bigg)^{y}\bigg(\frac{1}{\beta+1}\bigg)^{\alpha}
$
$ =\begin{pmatrix}y + \alpha -1 \\ y \end{pmatrix}  \left( \frac{\beta}{\beta+1}\right)^y \left( \frac{1}{\beta+1} \right)^\alpha$
