关于三角函数求不定积分的问题

∫（sinxcosx）/（sinx+cosx）dx
=∫（2sinxcosx）/[2√2sin(x+π/4)]dx
=∫sin2x/[2√2sin(x+π/4)]dx
=∫-cos(2x+π/2)/[2√2sin(x+π/4)]dx
=∫[2sin^2(x+π/4)-1]/[2√2sin(x+π/4)]dx
=∫｛√2/2*sin(x+π/4)+√2/4[-1/sin(x+π/4)]｝dx
=∫√2/2*sin(x+π/4)dx+∫√2/4[-1/sin(x+π/4)]dx
=-√2/2*cos(x+π/4)+√2/4*cos(x+π/4)/sin(x+π/4)+C