From the general definition of energy $E = \vec{F}\cdot\vec{s}$, together with equation $\vec{F} = m\vec{a}$, the kinetic energy or energy of motion of a body is:
\begin{equation}
E = \int_{s_1}^{s_2}m\vec{a}\cdot\mathrm{d}\vec{s}
\end{equation}
The mass of the body should not change, so it can be placed in front of the integral as a constant.
The acceleration can now be replaced by the derivative of the velocity with respect to time:
\begin{equation}
E = m\int_{s_1}^{s_2}\frac{\mathrm{d}\vec{v}}{\mathrm{d}t}\cdot\mathrm{d}\vec{s}
\end{equation}
Treating the differential quotient as a fraction yields the relationship $\vec{v} = \mathrm{d}\vec{s}/\mathrm{d}t$:
\begin{equation}
E = m\int_{v_1}^{v_2}\vec{v}\cdot\mathrm{d}\vec{v}
\end{equation}
From a mathematical perspective, variable substitution was performed here.
Multiplying out the scalar product and then integrating it finally yields the definition of kinetic energy:
\begin{equation}
E_{\mathrm{kin}} = \frac{1}{2}mv^2
\end{equation}
In this step, the lower limit was set to 0 and the upper limit to $v$.
This form of energy depends only on the mass $m$ and the magnitude of the final velocity $v$ reached by the body.
The exact dependence of acceleration on time is irrelevant for the result.