Answer
$$ \sin \left( {\ln x} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\cos \left( {\ln x} \right)}}{x}} dx \cr
& {\text{substitute }}u = \ln x,{\text{ }} \cr
& du = \frac{1}{x}dx \cr
& = \int {\cos \left( {\ln x} \right)\frac{1}{x}} dx \cr
& = \int {\cos u} du \cr
& {\text{find the antiderivative }} \cr
& = \sin u + C \cr
& {\text{write in terms of }}x,{\text{ replace }}u = \ln x \cr
& = \sin \left( {\ln x} \right) + C \cr} $$
