What's a better way to integrate an ODE for one time-step than the Euler method?
The next step is the trapezoidal rule; i.e., approximate the integrand with line segments between the sample points. If you draw it, you see that the correction is graphically the area of a rectangular triangle. Numerically one integration step gives an increment (delta t) x (the average of consequent samples).

The next step is to use Simpson's rule.

Numerical integration has been a must in science and engineering for hundreds of years. Before computers using effective methods was more important than today. Then the mathematicians also tried to estimate the error limit for the result. Their theories are still valid although lousy programming today only decreases the time increment until the result seems to stabilize.