How to finish some complex integration

Question

How to finish some integration as following below:

$$\int_x^{\infty} \frac{\mathrm \beta^{\alpha+\gamma} X^{\alpha-1}(y-x)^{\gamma-1}\exp^{-\beta y}}{\Gamma(\alpha) \Gamma(\gamma)}dy\;$$

and

$$\int_0^{y} \frac{\mathrm \beta^{\alpha+\gamma} X^{\alpha-1}(y-x)^{\gamma-1}\exp^{-\beta y}}{\Gamma(\alpha) \Gamma(\gamma)}dx\;$$

For your step by step answer will really appreciated.

Answer 1

For the first integral:

From the definition of the $\Gamma$-function and after substitutions you get

$$\int_x^\infty (y-x)^{\gamma-1} \cdot \exp(-\beta y) \, \mathrm{d}y= \exp(-x \cdot\beta)\cdot \beta^{-\gamma} \cdot \Gamma(\gamma)$$ when the real part of $\gamma$ and $\beta$ is strictly above $0$.

The substitutions should be $z=y-x$ from where you get the $\exp(-x\beta)$ part, and afterward s substitute to $u=\beta z$ from where you get $\beta^{-\gamma}$.