What is `int e^("In"(tan x)) dx` equal to ?

Options

A. In `|tan x| + c`
B. In `|sec x| + c`
C. `tan x + c`
D. `e^(tan x) + c`

Explanation

To solve the integral \( \int e^{\ln(\tan x)} \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
Using the property of exponents and logarithms, we know that \( e^{\ln(a)} = a \). Therefore, we can simplify the integrand:

\[ e^{\ln(\tan x)} = \tan x \]

### Step 2: Set up the integral
Now we can rewrite the integral:

\[ \int e^{\ln(\tan x)} \, dx = \int \tan x \, dx \]

### Step 3: Integrate \( \tan x \)
The integral of \( \tan x \) can be computed using the identity:

\[ \tan x = \frac{\sin x}{\cos x} \]

Thus, we can use the substitution method or recognize the integral directly. The integral of \( \tan x \) is:

\[ \int \tan x \, dx = -\ln|\cos x| + C \]

### Step 4: Rewrite the result
Using the property of logarithms, we can rewrite \( -\ln|\cos x| \) as:

\[ \int \tan x \, dx = \ln|\sec x| + C \]

### Final Answer
Thus, the final result of the integral is:

\[ \int e^{\ln(\tan x)} \, dx = \ln|\sec x| + C \]