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.