The Gaussian is your friend.

There is a special integral, highly bodacious, yet incredibly useful all over the place in mathematics and science: The Gaussian.

It comes disguised in all kinds of variations, and all of them are useful from time to time. Our textbook features a time-dependant external force F(t) in Figure 9.3 that looks like a Gaussian. The integral of it is discussed, and then approximated by the rectangle in Fig. 9.3(b).

So you might as well make your acquaintance with the Gaussian integral and start getting used to it. It looks heinous but it can be your friend.