1-2sin2(x) can be replaced by cos(2x), so the integrand is x2cos(2x).
Let u=x2, then du/dx=2x; let dv/dx=cos(2x), then v=½sin(2x).
Integrating by parts: uv-∫vdu=½x2sin(2x)-∫xsin(2x)dx.
Let u=x, then du/dx=1; let dv/dx=sin(2x), then v=-½cos(2x).
Integrating by parts: ∫xsin(2x)dx=-½xcos(2x)+½∫cos(2x)dx=
-½xcos(2x)+¼sin(2x)dx, therefore, ∫x2cos(2x)dx=½x2sin(2x)+½xcos(2x)-¼sin(2x)+C, where C is the integration constant.
CHECK
Differentiate ½x2sin(2x)+½xcos(2x)-¼sin(2x)+C:
x2cos(2x)+xsin(2x)-xsin(2x)+½cos(2x)-½cos(2x)=x2cos(2x)=x2(1-2sin2(x)).