∫sinx(cosx+1)dx/[1+(cosx)^2]
= -∫(1+cosx)dcosx/[1+(cosx)^2]
= -∫dcosx/[1+(cosx)^2] -(1/2)∫d[1+(cosx)^2]/[1+(cosx)^2]
= -arctan(cosx) - (1/2)ln[1+(cosx)^2] + C