How do I evaluate this gamma function integral?

Question

Is the following statement true? If so, can someone help me see why this is the case?

$\int_0^T x^k \exp(-x/x_0) dx = \Gamma(k + 1)x_0^k$

Answer 1

$$I=\int_0^T x^k\,e^{-\frac{x}{x_0}} dx=x_0^{(k+1)}\int_0^{\frac T{x_0}} t^k\,e^{-t}\,dt$$ $$I=x_0^{(k+1)} \Big[\Gamma (k+1)-\Gamma \left(k+1,\frac{T}{x_0}\right)\Big]$$