∫ (x + 2sinxcosx)/(1 + cos2x) dx
= ∫ x/(1 + cos2x) dx + ∫ 2sinxcosx/(1 + cos2x) dx
= ∫ x/(1 + 2cos²x - 1) dx + ∫ sin2x/(1 + cos2x) dx
= (1/2)∫ xsec²x dx - (1/2)∫ d(cos2x)/(1 + cos2x)
= (1/2)∫ x d(tanx) - (1/2)∫ d(1 + cos2x)/(1 + cos2x)
= (1/2)xtanx - (1/2)∫ tanx dx - (1/2)ln(1 + cos2x) + C
= (1/2)xtanx + (1/2)ln(cosx) - (1/2)ln(1 + cos2x) + C
= (1/2)xtanx + (1/2)ln(cosx) - (1/2)ln(1 + 2cos²x - 1) + C
= (1/2)[xtanx - ln(cosx)] + C