Question

Find value of \[\int {{e^{\log \left( {\tan x} \right)}}} \,dx\] is equal to
A. \[\log \,\,\tan \,x + c\]
B. \[\log \,\,\sec \,x + c\]
C. \[\tan \,x + c\]
D. \[{e^{\tan x}} + c\]

Answer 1

Hint: First apply logarithmic properties to simplify the equation. Then integrate the simplified version with respect to x. The value of integration is the required result in the question.

Complete step-by-step answer:
Given integration in the question can be written as:
\[\int {{e^{\log \left( {\tan x} \right)}}} \,dx\]
By assuming this integration to be I, we can write it as:
\[I = \int {{e^{\log \left( {\tan x} \right)}}} \,dx\]
By using basic properties of logarithm, we can say that:
\[{e^{\log x}} = x\]
By substituting the above equation, we can say the value as:
 \[I = \int {\tan x\,} \,dx\]
By basic integration properties, we know values of:
\[\int {\tan x\,} \,dx = \log \left( {\sec x} \right) + c\]
By substituting this into our equation, we can write it as:
\[I = \log \left( {\sec x} \right) + c\]
So the given integration can be simplified to \[\log \left( {\sec x} \right) + c\]
Therefore option (b) is correct for the given question.

Note: Be careful while writing the integration as every function is important. Whenever a log is with an empty base without loss of generality assume it as \[e\,\,{\log _e}\]. Here also we did the same that’s why we removed e with ease otherwise the solution would be very long to solve. The basic integrations must be remembered. It is very important to solve problems on integration.