\int f\left ( g\left (x\right )\right )g^\prime\left ( x \right )\hbox{d} x=\int  2x \sin x^2\hbox{d}x=\Bigg ( t:=x^2\Rightarrow \hbox{d}t=2x\hbox{d}x\Bigg)=\int \sin t\,\hbox{d}t=C-\cos t=C-\cos x^2