[Math] How to show that $f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$is a pdf

Question

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?

Answer 1

The following seems to be a typical definition of pdf for a basic probability course. I got it from this course site.

You will want to show that the function you have satisfies these conditions.