∫[(1+cos²x)/(1+cos2x)]dx

∫[(1+cos²x)/(1+cos2x)]dx

∫[(1+cos²x)/(1+cos2x)]dx=∫[(1+cos²x)/[cos²x+sin²x+cos²x-sin²x)]dx =1/2∫[(1+cos²x)/cos²x]dx=1/2∫sec²xdx+1/2∫dx =1/2tanx+(1/2)x+C.