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