The non-elementary functions
$$ F(x) = \int \sin(x^2)\mathrm dx $$ $$ G(x) = \int \cos(x^2)\mathrm dx $$
will yield
$$ F(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+3)}}{(2k+1)!(4k+3)}$$ $$ G(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+1)}}{(2k)!(4k+1)} $$
respectively, If you simply rewrite it to a Taylor series, and then integrate every term
However, some places (for example integral-calulator.com) claim that the integral is equal to
$$ F(x) = \sqrt{\frac{\pi}{2}} S\Big(\sqrt{\frac{2}{\pi}} x\Big) + C $$ $$ G(x) = \sqrt{\frac{\pi}{2}} C\Big(\sqrt{\frac{2}{\pi}} x\Big) + C $$ where $$S(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+3)}}{(2k+1)!(4k+3)} $$ $$C(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+1)}}{(2k)!(4k+1)} $$
Where does the $ \sqrt{\frac{\pi}{2}} $ and $ \sqrt{\frac{2}{\pi}}$ term come from? Why can't one just say that $ F(x) = S(x) + C $ and $ G(x) = C(x) + C $?