∫ 1/sinx dx = ∫ cscx dx

= ∫ cscx * (cscx - cotx)/(cscx - cotx) dx

= ∫ (- cscxcotx + csc²x)/(cscx - cotx) dx

= ∫ d(cscx - cotx)/(cscx - cotx)

= ln|cscx - cotx| + C

∫ 1/sinx dx

= ∫ 1/[2sin(x/2)cos(x/2)] dx

= ∫ 1/[cos²(x/2)tan(x/2)] d(x/2)

= ∫ 1/[tan(x/2)] d[tan(x/2)]

= ln|tan(x/2)| + C

∫ 1/sinx dx = ∫ sinx/sin²x dx

= ∫ 1/(cos²x - 1) d(cosx)

= (1/2)∫ [(cosx + 1) - (cosx - 1)]/[(cosx + 1)(cosx - 1)] d(cosx)

= (1/2)∫ [1/(cosx - 1) - 1/(cosx + 1)] d(cosx)

= (1/2)ln|(cosx - 1)/(cosx + 1)| + C

= (1/2)ln|[2sin²(x/2)]/[2cos²(x/2)]| + C

= (1/2) * 2ln|tan(x/2)| + C

= ln|tan(x/2)| + C

萬能代換：令y = tan(x/2)、dx = 2dy/(1 + y²)、sinx = 2y/(1 + y²)

∫ 1/sinx dx = ∫ 1/[2y/(1 + y²)] * 2dy/(1 + y²)

= ∫ (1 + y²)/(2y) * 2dy/(1 + y²)

= ∫ 1/y dy

= ln|y| + C

= ln|tan(x/2)| + C