Is it possible to intergrate sin(x^2) or cos(x^2)?
Hm, it's not infinite then? It's not like $\displaystyle sin(x^2)\rightarrow 0$ when $\displaystyle x\rightarrow \infty$, just a thought though, it's probably true if you say so.
By the way, I had more finite integrals in mind. There's not a primitive function or something?
Oh, I didn't realize it staibilizes because sinus and cosinus hovers between -1 and 1, the average will $\displaystyle \rightarrow$ 0.
But there can't be any primitive function, right? Cause I else I would have found a way to integrate the Gaussian function, and I guess it has been proven that there's no way to do it.
No there is not way to integrate the Gaussian integral. This is a famous result. However the integral $\displaystyle \int_{-\infty}^{\infty}e^{-x^2} dx = \sqrt{\pi}$ is a well-known and important result. The beauty of infinite integrals is that you can give a closed form answer though you cannot find the primitive.
There are many functions which do not have a closed form integral in terms
of elementary functions (in some sense these are more common that those
that do, but we don't tell you that, we only teach what can be done in
terms of elementary functions). The usual technique to deal with an
important class of integrals like this is to introduce a new class of special
functions in terms of which the integral can be written in cloded form.
An example which you may have met is the integral of the Gaussian density,
here we introduce the error function which allows us to write the integral in
closed form.
You might find this article of interest.
RonL