As i'm currently revising for my maths A-level i decided to put together a table of general results for integration of trig functions. I came across $$ \int cosec (kx)=-\frac 1k( \ln|cosec(kx)+cot(kx)|) dx +c $$ and $$ \int cosec (kx)=\frac 1k( \ln|tan(\frac12kx) dx +c $$
I was wondering if these general results are valid for all k? and when i plotted both the graphs of the first result and second result of this integral, the graphs overlapped slightly in radians but fully when i plotted them in degrees? i vaguely remember being told calc. trig dosent work in degrees In the exam can i use either result when i have to integrate cosec kx or similar? and furthermore
$$ \int sec (kx)=\frac 1k( \ln|sec(kx)+tan(kx)|) dx +c $$ and $$ \int sec (kx)=\frac 1k( \ln|tan(\frac 12kx+\frac 14 \pi)|) dx+c $$
for any question could i just pick either result to use? are they both valid for all k and for all x? (yes i know both can be achieved by manual integration each time but i'm making a general results table to memorise to speed up answering questions as running out of time is my main problem in maths)