*x*

^{2}) doesn't have an elementary antiderivative.

*Elementary*antiderivative? Clearly, they were hiding something. They didn't say it didn't have an antiderivative; they said it didn't have an

*elementary*antiderivative. Of course, I wanted to know what the antiderivative was. If it wasn't elementary, it had to be really awesome.

I looked up the integral of sin(

*x*

^{2}). Turns out, the integral cannot be expressed as anything other than itself. It's known as the Fresnel S integral, is written as S(

*x*), and is defined as the integral of sin(

*x*

^{2}). There's another Fresnel integral known as the Fresnel C integral which is written as C(

*x*) and defined as the integral of cos(

*x*

^{2}).

I also saw some graphs of the integrals. One really cool graph involved the parametric equations

*x*= C(

*t*) and

*y*= S(

*t*), and was called the "Euler spiral." It had a cool spirally shape, and I immediately knew that I had to graph it myself. I ended up writing an interactive JavaScript program to graph the parametric equations. Here it is; enjoy!

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