How to show that $f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$is a pdf? I have some problems trying to prove the following problem:

A continuous random variable $X$ is said to have a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$ if it has a pdf given by:
  $$f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$$ if $x>0$, or $0$ otherwise.

Given that apparently this is a pdf by definition, I do not know how to prove it is a pdf. My guess is to check if I take the integration of the distribution the value should be $1$. Is this correct? Is that enough?

$f(x)$ is a pdf if:
$f(x) \geq 0$ for all x. And,
$\int_{-\infty}^{\infty} f(x) dx = 1$
