Chapter 7 - Principles Of Integral Evaluation - 7.1 An Overview Of Integration Methods - Exercises Set 7.1 - Page 490: 8

$$\sec \left( {\ln x} \right) + C$$

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