$$
\begin{eqnarray*}
\frac{dx}{dt}(t_{n+1/2})&=&f(t_{n+1/2},x(t_{n+1/2}))\\
\class{eq1}{\frac{x(t_{n+1/2}+\Delta t/2)-x(t_{n+1/2}-\Delta t/2)}{\Delta t}}&\class{eq1}{\simeq} &\class{eq1}{f(t_{n+1/2},x(t_{n+1/2}))}\\
\class{eq2}{\frac{x(t_{n+1})-x(t_{n})}{\Delta t}}&\class{eq2}{\simeq} &\class{eq2}{f(t_{n+1/2},x(t_{n+1/2}))}\\
\class{eq3}{\frac{x_{n+1}-x_n}{\Delta t}}&\class{eq3}{=}&\class{eq3}{f(t_{n+1/2},x_{n+1/2})}\\
\class{eq4}{x_{n+1}}&\class{eq4}{=}&\class{eq4}{x_n+\Delta t f(t_{n+1/2},x_{n+1/2})}\\
\class{eq5}{x_{n+1}}&\class{eq5}{=}&\class{eq5}{x_n+\Delta t f(t_{n}+\Delta t/2,x_{n+1/2})}
\end{eqnarray*}
$$
$$
x_{n+1/2}=x_n+\frac{\Delta t}2 f(t_n,x_n)
$$
$$
\begin{eqnarray*}
x_{n+1/2}&=&x_n+\frac{\Delta t}2 f(t_n,x_n)\\
x_{n+1}&=&{x_n+\Delta t f(t_{n}+\Delta t/2,x_{n+1/2})}
\end{eqnarray*}
$$