Question

what is the integration of (tanx)/(1-cos x) the answer is (-ln|cos x|+cos x+c) but I don't know the steps :s

Answer 1

try this...
use your trig identities to transform the integrand, tanx/(1-cosx) to 1/(sinxcosx) + 1/sinx.
notice that 2/(2sinxcosx)=2/sin(2x) by the double angle formula. so in essence, you have \[\int\limits_{?}^{?}(tanx)/(1-cosx)dx=2\int\limits_{?}^{?}\csc(2x)dx+\int\limits_{?}^{?}\csc(x)dx\]

the answer to each of these integrals of csc is ln and you can write the sum/difference of lns as the ln of a product/quotient.