what is the$ \int \sin (x^2) \, dx$? $u$ substitution doesnt work. I don't see any connection with the Weierstrass substitution either. integration by parts results in a infinite integral series.

this integral does not have a solution in terms of elementary functions. But, you can solve it using series methods. For instance, since
$$ \sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$
$$ \text{then} \; \; \sin (x^2) = \sum \frac{(-1)^n x^{4n+2}}{(2n+1)!}$$
$$ \int \sin (x^2) = \int \sum \frac{(-1)^n x^{4n+2}}{(2n+1)!} = \sum \frac{(-1)^n x^{4n+3}}{(2n+1)!(4n+3)} + K $$
