Question

$\int e^{\log (\tan x) }dx$ is equal to :

• A. log tan x + c
• B. log sec x + c
• C. tan x + c
• D. $e^{\tan x} + c $

Answer 1

Answer: (B)

The given integral is
$I = \int e^{\log\left(\tan x\right)}dx = \int \tan x dx$
$ = \int\frac{\sin x}{\cos x} dx . \cos x =t$
$ \Rightarrow -\sin x dx = dt.$
So, $ I = -\int\frac{dt}{t} = - \log t +c = \log \frac{1}{t} + x$
$ = \log \frac{1}{\cos x} + c = \log\sec x + c $