Question

有道数分题帮忙做一下求大神帮忙做一下∫sinx/(1+sinx+cosx)dx=?急急急!？

Answer 1

∫ sinx/(1+sinx+cosx) dx
=∫sinx(sinx+cosx-1)/[(sinx+cosx+1)(sinx+cosx-1)] dx
=∫(sinxsinx+sinxcosx-sinx)/[(sinx+cosx)^2-1] dx
=∫(sinxsinx+sinxcosx-sinx)/(2sinxcosx) dx
=(1/2)∫sinx/cosx dx+(1/2)∫ dx-(1/2)∫1/cosx dx
=(-1/2)∫1/cosx d(cosx)+(1/2)∫ dx-(1/2)∫secx dx
=-(1/2)ln|cosx|+(1/2)x-(1/2)ln|secx+tanx|+C
说明：
∫sinx/cosxdx=-∫1/cosxdcosx=-ln|cosx|+c
∫secxdx=ln|secx+tanx|+C,8,2,