`inte^(-x)(1-tanx)secx dx` is equal to

Options

A. `e^(-x)sec x+C`
B. `e^(-x)tan x+C`
C. `-e^(-x)tanx+C`
D. none of these

Explanation

To solve the integral \( \int e^{-x} (1 - \tan x) \sec x \, dx \), we will follow a step-by-step approach.

### Step 1: Expand the Integral
We start by distributing \( \sec x \) in the integral:
\[ \int e^{-x} (1 - \tan x) \sec x \, dx = \int e^{-x} \sec x \, dx - \int e^{-x} \tan x \sec x \, dx \]

### Step 2: Rewrite the Integral
Now, we can rewrite the second integral:
\[ \int e^{-x} \sec x \, dx - \int e^{-x} \tan x \sec x \, dx \]

### Step 3: Recognize the Derivative
Notice that the derivative of \( \sec x \) is \( \tan x \sec x \). This allows us to use integration by parts or a specific formula for integrating products of exponential functions and trigonometric functions.

### Step 4: Use the Integration Formula
We can use the formula:
\[ \int e^{kx} (f(x) + f'(x)) \, dx = e^{kx} f(x) + C \]
In our case, \( k = -1 \) and \( f(x) = \sec x \), where \( f'(x) = \tan x \sec x \).

### Step 5: Apply the Formula
Applying the formula to our integral:
\[ \int e^{-x} \sec x \, dx - \int e^{-x} \tan x \sec x \, dx = e^{-x} \sec x + C \]

### Step 6: Combine the Results
Now, we combine the results:
\[ \int e^{-x} (1 - \tan x) \sec x \, dx = e^{-x} \sec x + C \]

### Step 7: Final Result
Thus, the final result of the integral is:
\[ -e^{-x} \sec x + C \]