Solve ∫ (from 0 to infty) of 1/betae^lambday-frac{y{beta wrt y

$$\int _ { 0 } ^ { \infty } \frac { 1 } { \beta } e ^ { \lambda y - \frac { y } { \beta } } d y$$

Evaluate

$$\left\{\begin{matrix}\frac{\infty}{\beta \lambda -1},&\left(\lambda =0\text{ and }\beta <0\right)\text{ or }\left(\beta \neq \frac{1}{\lambda }\text{ and }\beta <0\text{ and }\lambda >0\right)\text{ or }\left(\beta >\frac{1}{\lambda }\text{ and }\beta \neq 0\text{ and }\lambda >0\right)\text{ or }\left(\beta >\frac{1}{\lambda }\text{ and }\beta <0\text{ and }\lambda <0\right)\\-\frac{1}{\beta \lambda -1},&\left(\beta \neq \frac{1}{\lambda }\text{ and }\lambda <0\text{ and }\beta >0\right)\text{ or }\left(\beta \neq 0\text{ and }\beta <\frac{1}{\lambda }\text{ and }\lambda <0\right)\text{ or }\left(\beta >0\text{ and }\lambda =0\right)\text{ or }\left(\beta <\frac{1}{\lambda }\text{ and }\lambda >0\text{ and }\beta >0\right)\end{matrix}\right.$$

Differentiate w.r.t. β

$$\text{Indeterminate}$$