How do you find the integral of ##x^3 * sin( x^2 ) dx##?

##I=1/2(-x^2cosx^2+sinx^2)+C##

##x^2=t => 2xdx=dt, x^3dx=1/2x^2 2xdx = 1/2tdt##

##int x^3 sinx^2 dx = 1/2 int tsintdt = I##

##u=t => du=dt##

##dv=sintdt => v=int sintdt= -cost##

##I=1/2[-tcost + int costdt] = 1/2(-tcost+sint)+C##

##I=1/2(-x^2cosx^2+sinx^2)+C##