The graph of the antiderivative of f(x) = x e ^ (x)/(2) passes...

Let $g\left(x\right)$ represent the antiderivative of $f\left(x\right)$

Therefore, $g\left(x\right) = ∫ xe\dfrac{x}{2}dx$
$= x\left(2{e}^{\dfrac{x}{2}}\right) - ∫ \left(1\right) ⋅ 2{e}^{\dfrac{x}{2}}dx$…[integration by parts]
$= 2xe\dfrac{x}{2} - 4e\dfrac{x}{2} + c$
The graph of $y = g ( x )$ passes through $( 0, 3 )$
$⇒ 3 = 0 - 4{e}^{0} + c$
$⇒ c = 7$
Hence, $g(x) = 2x{e}^{\dfrac{x}{2}} - 4{e}^{\dfrac{x}{2}} + 7$
$⇒ g(x) = 2 ( x - 2 ) e\dfrac{x}{2} + 7$
$⇒ g ( 2 ) = 7, f ( 0 ) = 0$
$⇒ g ( 2 ) - f ( 0 ) = 7$