Correct Answer - Option 4 :
\(\rm \tan |\ln x| + C\)
Concept:
1. Integration by Substitution:
- If the given integration is of the form \(\smallint {\rm{g}}\left( {{\rm{f}}\left( {\rm{x}} \right)} \right){\rm{f'}}\left( {\rm{x}} \right){\rm{dx}}\) where \({\rm{g}}\left( {\rm{x}} \right)\) and \({\rm{f}}\left( {\rm{x}} \right)\) are both differentiable functions then we substitute \({\rm{f}}\left( {\rm{x}} \right) = {\rm{u}}\) which implies that \({\rm{f'}}\left( {\rm{x}} \right){\rm{dx}} = {\rm{du}}\).
- Therefore, the integral becomes \(\smallint {\rm{g}}\left( {\rm{u}} \right){\rm{du}}\) which can be solved by general formulas.
- \(\rm \smallint \sec^2 x\cdot dx = \tan x + c\)
Solution:
Given: \(\rm \smallint \frac {\sec^2 ({ln{(x)}})}{x} dx\)
In the given problem substitute \(\ln \left( {\rm{x}} \right) = u\) therefore, \(\frac{{{\rm{dx}}}}{{\rm{x}}} = {\rm{du}}\).
The given integral becomes
\(\smallint \sec^2 {\rm{u}}\cdot du= \tan {\rm{u}} + {\rm{C}}\)
Resubstitute \({\rm{u}} = \ln \left( {\rm{x}} \right)\).
\(\rm \smallint \frac {\sec^2 ({ln{(x)}})}{x} dx = \tan |\ln x| + C\)
