Integrate by parts. Let u=θ2 and dv=sin(2θ)dθ, so:
du=2θdθ, v=-cos(2θ)/2.
∫udv=uv-∫vdu=-θ2cos(2θ)/2+∫θcos(2θ)dθ.
∫θcos(2θ)dθ by parts: let u=θ, du=dθ; dv=cos(2θ)dθ, v=sin(2θ)/2.
∫udv=θsin(2θ)/2-½∫sin(2θ)dθ=θsin(2θ)/2+¼cos(2θ).
Complete integral is:
-θ2cos(2θ)/2+θsin(2θ)/2+¼cos(2θ)+C where C is integration constant.
CHECK
Differentiate -θ2cos(2θ)/2+θsin(2θ)/2+¼cos(2θ)+C:
-θcos(2θ)+θ2sin(2θ)+½sin(2θ)+θcos(2θ)-½sin(2θ)=θ2sin(2θ). Checks out OK.