If `f(x)=int(2+sqrt(x))/((x+1+sqrt(x))^2)dx` and `f(0)=0` then the value of If [f(4)] is: (where [.] represents the greatest integer function) .

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to evaluate the integral function given by:

\[ f(x) = \int \frac{2 + \sqrt{x}}{(x + 1 + \sqrt{x})^2} \, dx \]

and find \( f(4) \) such that \( f(0) = 0 \), and then determine \([f(4)]\), where \([.]\) denotes the greatest integer function.

### Step 1: Substitution
Let us make the substitution \( \sqrt{x} = t \). Then, we have:

\[ x = t^2 \quad \text{and} \quad dx = 2t \, dt \]

Now we can rewrite the integral:

\[ f(x) = \int \frac{2 + t}{(t^2 + 1 + t)^2} \cdot 2t \, dt \]

### Step 2: Simplifying the Integral
The integral now becomes:

\[ f(t) = 2 \int \frac{(2 + t)t}{(t^2 + t + 1)^2} \, dt \]

### Step 3: Differentiation
To simplify the integral further, we can differentiate the function \( \frac{t + 1}{t^2 + t + 1} \):

Let \( u = t + 1 \) and \( v = t^2 + t + 1 \). We find:

\[ \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]

Calculating \( u' = 1 \) and \( v' = 2t + 1 \):

\[ \frac{d}{dt} \left( \frac{t + 1}{t^2 + t + 1} \right) = \frac{(t^2 + t + 1)(1) - (t + 1)(2t + 1)}{(t^2 + t + 1)^2} \]

### Step 4: Evaluating the Integral
After simplifying, we can express \( f(t) \):

\[ f(t) = 2 \left( -\frac{t + 1}{t^2 + t + 1} + C \right) \]

Substituting back \( t = \sqrt{x} \):

\[ f(x) = -2 \frac{\sqrt{x} + 1}{x + 1 + \sqrt{x}} + C \]

### Step 5: Finding the Constant \( C \)
Given \( f(0) = 0 \):

\[ f(0) = -2 \frac{0 + 1}{0 + 1 + 0} + C = -2 + C = 0 \]

Thus, \( C = 2 \).

### Step 6: Final Expression for \( f(x) \)
Now we can write the final expression for \( f(x) \):

\[ f(x) = -2 \frac{\sqrt{x} + 1}{x + 1 + \sqrt{x}} + 2 \]

### Step 7: Evaluating \( f(4) \)
Now we compute \( f(4) \):

\[ f(4) = -2 \frac{2 + 1}{4 + 1 + 2} + 2 = -2 \frac{3}{7} + 2 = -\frac{6}{7} + 2 = 2 - \frac{6}{7} = \frac{14}{7} - \frac{6}{7} = \frac{8}{7} \]

### Step 8: Finding the Greatest Integer Function
Now we need to find \([f(4)]\):

\[ [f(4)] = \left[\frac{8}{7}\right] = 1 \]

### Final Answer
Thus, the value of \([f(4)]\) is:

\[ \boxed{1} \]