Answer
$$ - \ln \left| {\cos \left( {\ln v} \right)} \right| + C $$
Work Step by Step
$$\eqalign{
& \int {\frac{{\tan \left( {\ln v} \right)}}{v}} dv \cr
& {\text{Integrate by the substitution method}} \cr
& {\text{set }}u = \ln v{\text{ then }}\frac{{du}}{{dv}} = \frac{1}{v},\,\,\,\,dv = vdu \cr
& {\text{write the integrand in terms of }}u \cr
& \int {\frac{{\tan \left( {\ln v} \right)}}{v}} dv = \int {\frac{{\tan u}}{v}} \left( {vdu} \right) \cr
& {\text{cancel common terms}} \cr
& = \int {\tan u} du \cr
& {\text{integrating}}{\text{, we use }}\cr
& \int {\tan x} dx = - \ln \left| {\cos x} \right| + C \cr
& = - \ln \left| {\cos u} \right| + C \cr
& {\text{replace }}\ln v{\text{ for }}u \cr
& = - \ln \left| {\cos \left( {\ln v} \right)} \right| + C \cr} $$
