Find y= f(x) with intergrals?

if $\displaystyle f "(x)= 2x-1$, f ' (1) = 3 and f(0)= 1

Okay for this one, for the first integral I got $\displaystyle x^2 -x+c$. I tested out the F'(1) = 3 and got 0+C=3

Then for the 2nd integral I got $\displaystyle 1/3 x^3 - 1/2 x^2 +c$. I tested out f(0) = 1 and got 0-0+c=1

Correct?