This is a definite integral. There is no added constant
$\begin{align*}
&\displaystyle \int_2^x (5t^2 - 4t)~dt = \\
&\left . \dfrac 5 3 t^3 - 2t^2 \right |_2^x = \\
&\left(\dfrac 5 3 x^3 - 2 x^2\right) - \left(\dfrac 5 3 2^3 - 2(2)^2\right) = \\
&\left(\dfrac 5 3 x^3 - 2 x^2\right) - \left(\dfrac{40}{3} - 8\right ) = \\
&\dfrac 5 3 x^3 - 2 x^2 - \dfrac{16}{3}
\end{align*}$