`int ((2+sqrtx)dx )/((x+1+sqrtx)^(2))` is equal to:

Options

A. `(x)/(x+sqrtx+1)+C`
B. `(2x)/(x+sqrtx+1)+C`
C. `(-2x)/(x+sqrtx+1)+C`
D. `(-x)/(x+sqrtx+1)+C`

Answer

To solve the integral

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

we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral as given:

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

### Step 2: Substitute for Simplicity
Let us make the substitution:

\[ t = x + 1 + \sqrt{x}. \]

To differentiate, we need to express \( \sqrt{x} \) in terms of \( t \):

\[ \sqrt{x} = t - x - 1. \]

Now, differentiating both sides with respect to \( x \):

\[ \frac{1}{2\sqrt{x}} \, dx = dt - dx. \]

Rearranging gives us:

\[ dx = 2\sqrt{x} \, dt. \]

### Step 3: Express \( \sqrt{x} \) in terms of \( t \)
From our substitution, we can express \( \sqrt{x} \) as:

\[ \sqrt{x} = t - x - 1. \]

### Step 4: Substitute Back into the Integral
Now, substituting back into the integral, we have:

\[ \int \frac{2 + (t - x - 1)}{t^2} \cdot 2\sqrt{x} \, dt. \]

### Step 5: Simplify the Integral
This simplifies to:

\[ \int \frac{1 + t - x}{t^2} \cdot 2\sqrt{x} \, dt. \]

### Step 6: Change of Variables
Now, we need to express everything in terms of \( t \). We can express \( x \) in terms of \( t \) and \( \sqrt{x} \):

\[ x = (t - 1)^2. \]

### Step 7: Final Integral
Substituting these back into the integral, we can simplify and integrate:

\[ \int \frac{2}{t^2} \, dt. \]

### Step 8: Integrate
The integral of \( \frac{2}{t^2} \) is:

\[ - \frac{2}{t} + C. \]

### Step 9: Substitute Back for \( t \)
Substituting back for \( t \):

\[ - \frac{2}{x + 1 + \sqrt{x}} + C. \]

### Final Answer
Thus, the solution to the integral is:

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

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