A distribution X is described by the probability density function
$f(x) = \frac{x}{\alpha^2}e^{-\frac{x^2}{2\alpha^2}}$, where $x\ge0.$
Find the cumulative distribution function.
What I have so far is $F(x) = 1-e^{-\frac{x^2}{2\alpha^2}}$ for $x\ge0$ using the integral $\int^{x}_0\frac{t}{\alpha^2}e^{-\frac{t^2}{2\alpha^2}}dt$ .
Is this all I need? I feel like I'm missing something...
Assuming $\alpha \neq 0$, your answer is correct. (If $\alpha = 0$, the PDF is undefined, so there's not much to do there either.)