# Improve the accuracy of $F(t) = \int\limits_0^t f(t - \tau) d\tau$

I'd like to integrate the following very simple integral

$$F(t) = \int\limits_0^t f(t - \tau) d\tau$$

Here, let's assume that $$f(t)$$ is any function, $$t$$ is in domain $$[0, 1]$$ while the sub-interval is $$dt=0.1$$ and $$0<\tau < t$$.

We can numerically integrate it by using the Trapezoidal rule when $$t=0.1$$, and Simpson's one when $$t=0.2, 1.0$$, Simpson's 3/8 one when $$t=0.2, 0.6, 0.9$$ and Boole's one when $$t=0.4, 0.8$$. Of course, we can just use Trapezoid one over the domain. However, we know that the total error of the numerical results mainly depend on the low accuracy of Trapezoid rule.

Finally, do you have any idea for improve the integration not using the Trapezoid one?

Looking forward to hearing from you.

Thanks:)

• By the way, the integral is the same as $\int_0^t f(\tau)\, d\tau$. Use a substitution $u = t-\tau$ to see this. – Minus One-Twelfth Apr 6 at 8:47