If `f(x) = (1)/(cos^2xsqrt(1-tan x))` then its anti-derivative 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+tanx)+ 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 \( F(0) = 4 \), we will follow these steps:

### Step 1: Set up the integral
We start with the function:
\[ F(x) = \int f(x) \, dx = \int \frac{1}{\cos^2 x \sqrt{1 - \tan x}} \, dx \]

**Hint:** Remember that \( \tan x = \frac{\sin x}{\cos x} \) and \( \cos^2 x = \frac{1}{\sec^2 x} \).

### Step 2: Substitute for \( \sqrt{1 - \tan x} \)
Let:
\[ t = \sqrt{1 - \tan x} \]
Then, differentiate both sides:
\[ \frac{dt}{dx} = \frac{1}{2\sqrt{1 - \tan x}} \cdot (-\sec^2 x) = -\frac{\sec^2 x}{2\sqrt{1 - \tan x}} \]
Thus, we can express \( dx \) in terms of \( dt \):
\[ dx = -2 \frac{\sqrt{1 - \tan x}}{\sec^2 x} dt \]

**Hint:** Use the relationship between \( \tan x \) and \( \sec^2 x \) to simplify further.

### Step 3: Substitute into the integral
Now substitute \( dx \) into the integral:
\[ F(x) = \int \frac{1}{\cos^2 x \sqrt{1 - \tan x}} \left(-2 \frac{\sqrt{1 - \tan x}}{\sec^2 x}\right) dt \]
This simplifies to:
\[ F(x) = -2 \int dt \]

**Hint:** Remember that \( \sec^2 x = \frac{1}{\cos^2 x} \).

### Step 4: Integrate
The integral of \( dt \) is:
\[ F(x) = -2t + C \]

**Hint:** Don't forget to substitute back for \( t \).

### Step 5: Substitute back for \( t \)
Substituting back \( t = \sqrt{1 - \tan x} \):
\[ F(x) = -2\sqrt{1 - \tan x} + C \]

### Step 6: Use the initial condition
We have the condition \( F(0) = 4 \). First, calculate \( F(0) \):
\[ F(0) = -2\sqrt{1 - \tan(0)} + C = -2\sqrt{1 - 0} + C = -2 + C \]
Setting this equal to 4 gives:
\[ -2 + C = 4 \implies C = 6 \]

**Hint:** Always check your calculations when substituting values.

### Step 7: Write the final answer
Thus, the anti-derivative is:
\[ F(x) = -2\sqrt{1 - \tan x} + 6 \]

### Final Answer:
\[ F(x) = -2\sqrt{1 - \tan x} + 6 \]