determine distribution function and nth moment

Hi, I'm so struggling with this quetsion, can anyone please kindly help me? Thanks a lot.

Let $\displaystyle \alpha, \beta >0$ and

$\displaystyle f(x)=\left\{\begin{array}{cc}\frac{\alpha}{\beta}( \frac{x}{\alpha})^{\alpha -1} \exp(-(\frac{x}{\beta})^\alpha),&\mbox{ if } x>0\\0,&\mbox{otherwise}\end{array}\right$

and suppose the random variable X has density f.

(a)Determine the distribution function $\displaystyle F(x)$ of X

(b)Calculate the nth moment $\displaystyle E(X^n)$ for all n=1,... Express it in terms of the Gamma function.

I did try to integrate PDF to get distribution function, but I end up a constant $\displaystyle \frac{\alpha}{\beta}$, it must be wrong. Also, I have no idea about the second one. Can anyone please help me? Thanks a lot.