integration - Simple explanation of difference between $cos(x)$ and $cos(x^2)$ integrals.

I'm not looking for the solved integrals, I'm just looking for a simple explanation for why

$\displaystyle\int\cos(x^2)dx$

is so much more complicated than

$\displaystyle\int\cos(x)dx$

With simple I mean something I can say to explain the difference to a person just starting to learn about integration. Perhaps something visual?

Answer

For one thing, look at their graphs. Since $x$ increases at a constant rate, $\cos(x)$ has a constant period but $x^2$ increases faster and faster for larger $x$ so the "period" of $\cos(x^2)$ keeps getting smaller and smaller.