Duhamel's integral

Duhamel's Integral is a fancy way of figuring out what happens to something over time when it's being affected by something else. Imagine you're playing with a toy car and you want to know how it moves if you push it while it's already moving.

To do this, you need to figure out how the car would have moved if you hadn't pushed it at all, and then add on the extra movement caused by your push. Duhamel's Integral helps you do this by breaking down the movement into tiny pieces and figuring out how each piece would have moved on its own.

It's like when you're putting together a puzzle and you have to look at each individual piece to figure out where it goes. Duhamel's Integral lets you look at each tiny piece of the car's movement and figure out how the push affected it.

It might sound complicated, but it's really just a math equation that helps you solve this problem. And scientists and engineers use it all the time to understand how things work in the world around us.