Question

1/（1-cosx）的不定积分 求～

Answer 1

∫ 1/(1 - cosx) dx
= ∫ (1 + cosx)/[(1 - cosx)(1 + cosx)] dx
= ∫ (1 + cosx)/(1 - cos^2(x)) dx
= ∫ (1 + cosx)/sin^2(x) dx
= ∫ (csc^2(x) + cscxcotx) dx
= - cotx - cscx + C
或
∫ 1/(1 - cosx) dx
= ∫ 1/[2sin^2(x/2)] dx
= ∫ csc^2(x/2) d(x/2)
= - cot(x/2) + C

Answer 2

1-cosx=2sin²(x/2)
1/（1-cosx）=0.5csc²（x/2）

∫dx/(1-cosx)
=∫0.5csc²(x/2)dx
=∫sec²(x/2)d0.5x
=-∫dcot(x/2)
=-cot(x/2)+c