`int1/(2sqrt((x)) (1 + x)) dx` =

Options

A. `"cot"^(-1( sqrt(x)`
B. `"tan"^(-1) sqrt(x)`
C. `"sec"^(-1) sqrt(x)`
D. none

Answer

To solve the integral \( \int \frac{1}{2\sqrt{x}(1+x)} \, dx \), we will follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ \int \frac{1}{2\sqrt{x}(1+x)} \, dx \]

### Step 2: Substitute \( t = \sqrt{x} \)
Let’s make the substitution \( t = \sqrt{x} \). Then, we have:
\[ x = t^2 \quad \text{and} \quad dx = 2t \, dt \]

### Step 3: Substitute into the integral
Now we substitute \( \sqrt{x} \) and \( dx \) in the integral:
\[ \int \frac{1}{2t(1+t^2)} (2t \, dt) = \int \frac{1}{1+t^2} \, dt \]

### Step 4: Integrate
The integral \( \int \frac{1}{1+t^2} \, dt \) is a standard integral that equals:
\[ \tan^{-1}(t) + C \]

### Step 5: Substitute back \( t = \sqrt{x} \)
Now we substitute back \( t \) to get the result in terms of \( x \):
\[ \tan^{-1}(\sqrt{x}) + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{1}{2\sqrt{x}(1+x)} \, dx = \tan^{-1}(\sqrt{x}) + C \]

---