what is the integral of (1-cos[x])/(sin[x])?

Answer 1

Find the integral of [1-cos(x)]/sin(x) with respect to x.

First break apart the fraction then integrate.

[1-cos(x)]/sin(x) = 1/sin(x) - cos(x)/sin(x) = csc(x) - cot(x)

Now we can integrate.

∫{[1-cos(x)]/sin(x)}dx = ∫{csc(x) - cot(x)}dx
= -ln|csc(x) + cot(x)| - ln|sin(x)| + C

Answer 2

this seems simple enough, why don't you first divide them into parts so we get
1/sinx - cosx/sinx = secx - cotx

Then you can integrate secx and cotx by itself, and hopefully if you know standard integrals (http://metric.ma.ic.ac.uk/integration/techniques/indefinite/standard-integrals/index.html),

then you would know that the integral of secx = ln (secx + tanx) +c

and the integral of cotx = ln(sinx) +c

Therefore putting the integral would be: ln(secx + tanx) - ln(sinx) +c