If `f(x) =(1)/(cos^(2)xsqrt(1+tanx))`, then its anit-derivate F(x) satisfying F(0) = 4 is :

Options

A. `sqrt(1+ tan x) + 4`
B. `(2)/(3) (1+ tan x)^(3//2)`
C. `2(sqrt(1+ tan x) + 1)`
D. None of these

Answer

To find the anti-derivative \( F(x) \) of the function \( f(x) = \frac{1}{\cos^2 x \sqrt{1 + \tan x}} \) that satisfies the condition \( F(0) = 4 \), we will follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = \frac{1}{\cos^2 x \sqrt{1 + \tan x}} \]

### Step 2: Substitute for simplification
Let:
\[ t = \sqrt{1 + \tan x} \]
Then, we need to find \( dt \) in terms of \( dx \).

### Step 3: Differentiate the substitution
Differentiating both sides with respect to \( x \):
\[ dt = \frac{1}{2\sqrt{1 + \tan x}} \cdot \sec^2 x \, dx \]
This implies:
\[ dx = \frac{2\sqrt{1 + \tan x}}{\sec^2 x} dt \]

### Step 4: Substitute \( dx \) back into the integral
Now, substituting \( dx \) back into the integral:
\[ \int f(x) \, dx = \int \frac{1}{\cos^2 x \sqrt{1 + \tan x}} \cdot \frac{2\sqrt{1 + \tan x}}{\sec^2 x} dt \]

### Step 5: Simplify the integral
Notice that:
\[ \sec^2 x = \frac{1}{\cos^2 x} \]
Thus, we can simplify:
\[ \int f(x) \, dx = \int 2 \, dt = 2t + C \]

### Step 6: Substitute back for \( t \)
Now substitute back \( t = \sqrt{1 + \tan x} \):
\[ F(x) = 2\sqrt{1 + \tan x} + C \]

### Step 7: Use the initial condition to find \( C \)
We know that \( F(0) = 4 \). First, calculate \( \tan(0) \):
\[ F(0) = 2\sqrt{1 + \tan(0)} + C = 2\sqrt{1 + 0} + C = 2 + C \]
Setting this equal to 4:
\[ 2 + C = 4 \implies C = 2 \]

### Step 8: Write the final expression for \( F(x) \)
Thus, the anti-derivative \( F(x) \) is:
\[ F(x) = 2\sqrt{1 + \tan x} + 2 \]

### Final Answer
The anti-derivative \( F(x) \) satisfying \( F(0) = 4 \) is:
\[ F(x) = 2\sqrt{1 + \tan x} + 2 \]