求下列不定积分 ∫1/[sin(2x)+2sin x]dx

求下列不定积分 ∫1/[sin(2x)+2sin x]dx
求下列不定积分 ∫1/[sin(2x)+2sin x]dx

求下列不定积分 ∫1/[sin(2x)+2sin x]dx
∫1/[sin(2x)+2sin x]dx
=∫1/[2sinxcosx+2sin x]dx
=∫1/(2sinx*[cosx+1]) dx
=∫1/(sinx*[2cos^2(x/2)]) d(x/2)
=1/2∫1/sinx dtan(x/2)
=1/2∫[1+tan^2(x/2)]/[2tan(x/2)] dtan(x/2)
=1/4∫[1/tan(x/2) ]+tan(x/2) dtan(x/2)
=1/4 ln|tan(x/2)|+1/8tan^2(x/2) +C