Use a derivative inside integration by parts

$\dfrac d{dx}\sin(x^2)=\cos(x^2)\cdot2x=2x\cos(x^2)$.

Write the integral as $\dfrac12\int x^2\cdot2x\cos(x^2)\,dx$. Use u=x², dv/dx=2x cos(x²), du/dx=2x and v=sin(x²). Parts gives $\dfrac12[x^2\sin(x^2)-\int2x\sin(x^2)\,dx]$. The remaining integral is −cos(x²), so the result is $x^2\sin(x^2)/2+\cos(x^2)/2+C$.