The value of `int (tan x )/(tan ^(2) x + tan x+1)dx =x -(2)/(sqrtA) tan ^(-1) ((2 tan x+1)/(sqrtA))+C` Then the value of A is:

Answer

To solve the integral \( I = \int \frac{\tan x}{\tan^2 x + \tan x + 1} \, dx \) and find the value of \( A \) such that the result matches the given expression, we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\tan x}{\tan^2 x + \tan x + 1} \, dx \]
To simplify the denominator, we can rewrite it by adding and subtracting 1:
\[ \tan^2 x + \tan x + 1 = \tan^2 x + \tan x + 1 - 1 + 1 = (\tan^2 x + \tan x + 1) + 1 - 1 \]
This allows us to express the integral in a more manageable form.

### Step 2: Split the Integral
Now we can express the integral as:
\[ I = \int \left( \frac{\tan x + 1 - 1}{\tan^2 x + \tan x + 1} \right) \, dx \]
This can be split into two parts:
\[ I = \int \frac{\tan x + 1}{\tan^2 x + \tan x + 1} \, dx - \int \frac{1}{\tan^2 x + \tan x + 1} \, dx \]

### Step 3: Substitution
Let \( t = \tan x \). Then, \( dt = \sec^2 x \, dx \) or \( dx = \frac{dt}{\sec^2 x} = \frac{dt}{1 + t^2} \).

### Step 4: Change of Variables
Substituting \( t \) into the integral:
\[ I = \int \frac{t}{t^2 + t + 1} \cdot \frac{dt}{1 + t^2} \]
This simplifies to:
\[ I = \int \frac{t}{(t^2 + t + 1)(1 + t^2)} \, dt \]

### Step 5: Further Simplification
We can simplify the denominator:
\[ t^2 + t + 1 = (t + \frac{1}{2})^2 + \frac{3}{4} \]
This leads us to a form suitable for integration.

### Step 6: Integration
Using the standard integral formula:
\[ \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C \]
we can integrate the expression.

### Step 7: Combine Results
After performing the integration and simplifying, we find:
\[ I = x - \frac{2}{\sqrt{3}} \tan^{-1} \left( \frac{2 \tan x + 1}{\sqrt{3}} \right) + C \]

### Step 8: Compare with Given Expression
We compare this result with the given expression:
\[ x - \frac{2}{\sqrt{A}} \tan^{-1} \left( \frac{2 \tan x + 1}{\sqrt{A}} \right) + C \]
From this comparison, we can see that:
\[ \frac{2}{\sqrt{3}} = \frac{2}{\sqrt{A}} \]
Squaring both sides gives:
\[ 3 = A \]

### Final Answer
Thus, the value of \( A \) is:
\[ \boxed{3} \]