Интеграл
u=x+2π
du=(x+2π)`dx
du=dx
dv=sin2xdx
v= ∫ dv= ∫ sin2xdx=(1/2) ∫ sin2x(2x)=(1/2)*(-cos2x)
[b]∫ udv=u*v- ∫ vdu[/b]
∫ ^π_(0)(x+2π)*sin2x*dx=
=(x+2π)*(1/2)*(-cos2x)|^π_(0)- ∫ ^π_(0)(1/2)*(-cos2x)dx=
= (π+2π)*(1/2)(-cos4π)-(0+2π)*(1/2)*(-cos0) + (1/4)∫ ^π_(0)cos2xd(2x)=
=-(3/2)π+π +(1/4)*(sin2x)|^π_(0) = (-π/2) +(1/4)sin2π-(1/4)sin0=
[b]=-π/2[/b]