I googled and remembered that it's a section of the Euler curve, which is a curve who's curvature increases linearly along its length.

So the clothoid loop is the portion of the Euler curve that moves from the origin to its highest point, then mirrored so that it comes back down. It's like taking a turn in your car and turning the steering wheel at a constant rate.

I can't remember the exact way to form the integral or what it's used for. But I do know that you can't solve the integral analytically.