Integral of $$$u^{\alpha - 2} e^{- u}$$$ with respect to $$$u$$$

Find $$$\int u^{\alpha - 2} e^{- u}\, du$$$.

This integral (Incomplete Gamma Function) does not have a closed form:

$${\color{red}{\int{u^{\alpha - 2} e^{- u} d u}}} = {\color{red}{\left(- \Gamma\left(\alpha - 1, u\right)\right)}}$$

Therefore,

$$\int{u^{\alpha - 2} e^{- u} d u} = - \Gamma\left(\alpha - 1, u\right)$$

Add the constant of integration:

$$\int{u^{\alpha - 2} e^{- u} d u} = - \Gamma\left(\alpha - 1, u\right)+C$$

$$$\int u^{\alpha - 2} e^{- u}\, du = - \Gamma\left(\alpha - 1, u\right) + C$$$A