I have to find the marginal density of X. Then, I need to find the variance of X.
My attempt: I think the marginal density is just the integral, which is 2x from 0 < x < 1
How do I find the variance? I know there is this formula that uses the expected value, but I'm not sure how to proceed: Var(X) = E([X - E(X)]^2 )
Maybe something like E([2X - 1]^2)?
Your pdf of $X$ is right. The variance can be calculated as
$Var(X)=E(X^2)-[E(X)]^2$
In your case:
$Var(X)=\int_0^1 x^2\cdot f_X(x) \, dx-\left[\int_0^1 x\cdot f_X(x) \, dx\right]^2$
$Var(X)=\int_0^1 x^2\cdot 2x \, dx-\left[\int_0^1 x\cdot 2x \, dx\right]^2$