`"If " int sinx d(secx)=f(x)-g(x)+c,` then

Options

A. `f(x)=secx`
B. `f(x)=tanx`
C. `g(x)=2x`
D. `g(x)=x`

Explanation

To solve the integral \( \int \sin x \, d(\sec x) = f(x) - g(x) + c \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \sin x \, d(\sec x) \]
To proceed, we multiply and divide by \( dx \):
\[ I = \int \sin x \, \frac{d(\sec x)}{dx} \, dx \]

### Step 2: Differentiate \( \sec x \)
The derivative of \( \sec x \) is:
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \]
Thus, we can rewrite the integral as:
\[ I = \int \sin x \cdot \sec x \tan x \, dx \]

### Step 3: Rewrite \( \sin x \) in Terms of \( \tan x \)
We know that:
\[ \sin x = \frac{\tan x}{\sqrt{1 + \tan^2 x}} = \frac{\tan x}{\sec x} \]
Substituting this into the integral gives:
\[ I = \int \frac{\tan x}{\sec x} \cdot \sec x \tan x \, dx = \int \tan^2 x \, dx \]

### Step 4: Use the Identity for \( \tan^2 x \)
Using the identity:
\[ \tan^2 x = \sec^2 x - 1 \]
We can rewrite the integral:
\[ I = \int (\sec^2 x - 1) \, dx \]

### Step 5: Integrate
Now we can integrate:
\[ I = \int \sec^2 x \, dx - \int 1 \, dx \]
The integral of \( \sec^2 x \) is \( \tan x \) and the integral of \( 1 \) is \( x \):
\[ I = \tan x - x + C \]

### Step 6: Identify \( f(x) \) and \( g(x) \)
From our result, we can identify:
\[ f(x) = \tan x \quad \text{and} \quad g(x) = x \]

### Final Answer
Thus, the final answer is:
\[ \int \sin x \, d(\sec x) = \tan x - x + C \]