Question

what is the anit-derivative of tanx

Answer 1

ln sec x

Answer 2

\[-\ln (\left| cosx \right|)+C\]or\[\ln (\left| secx \right|)+C\]You just have to memorize trig integrals like this, or you may also be able to find it by using integration by parts.

Answer 3

Actually I just realized with this one you do this:\[\int\limits_{}^{}(tanx) dx\]\[\int\limits_{}^{}(sinx/cosx)dx\]\[-\int\limits_{}^{}(-sinx/cosx)dx\]Now use substitution where u=cosx and du=-sinx
\[-\int\limits_{}^{}(1/u)du\]\[-\ln \left| u \right|+C\]\[-\ln \left| cosx \right|+C\]