Let $f(x)$ be a function satisfying $f'(x) = f(x)$ with $f(0) = 1$ and $g(x)$ be a function that satisfies $f(x) + g(x) = x^2$. Then the value of the integral $\begin{align*} \int_0^1 f(x) \; g(x) \;dx \end{align*} $ is :

Question

( A ) $e + \frac{e^2}{2} - \frac{3}{2}$
( B ) $e - \frac{e^2}{2} - \frac{5}{2}$
( C ) $e + \frac{e^2}{2} + \frac{5}{2}$
( D ) $e - \frac{e^2}{2} - \frac{3}{2}$