∫ (x+sinx)/(1+cosx) dx
=∫ x/(1+cosx) dx+∫ sinx/(1+cosx) dx
=∫ x/(1+cosx) dx+∫ 2sin(x/2)cos(x/2)/(2cos²(x/2)) dx
=∫ x/(2cos²(x/2)) dx+∫ tan(x/2) dx
=∫ xsec²(x/2) d(x/2)+∫ tan(x/2) dx
=∫ xd(tan(x/2))+∫ tan(x/2) dx
=xtan(x/2)-∫ tan(x/2) dx+∫ tan(x/2) dx
=xtan(x/2)+C