Evaluate the following Integrals :
`int (dx)/((x+1)^(1//2)+(x+1)^(1//2))`

Answer

To evaluate the integral

\[ I = \int \frac{dx}{\sqrt{x+1} + \sqrt{x+1}}, \]

we can simplify the expression inside the integral first.

### Step 1: Simplify the integrand
Notice that \(\sqrt{x+1} + \sqrt{x+1} = 2\sqrt{x+1}\). Therefore, we can rewrite the integral as:

\[ I = \int \frac{dx}{2\sqrt{x+1}}. \]

### Step 2: Factor out the constant
We can factor out the constant \( \frac{1}{2} \) from the integral:

\[ I = \frac{1}{2} \int \frac{dx}{\sqrt{x+1}}. \]

### Step 3: Use substitution
Now, let's use the substitution \( t = x + 1 \). This means that \( dx = dt \). When \( x = 0 \), \( t = 1 \). Thus, the integral becomes:

\[ I = \frac{1}{2} \int \frac{dt}{\sqrt{t}}. \]

### Step 4: Integrate
The integral of \(\frac{1}{\sqrt{t}}\) is \(2\sqrt{t}\):

\[ I = \frac{1}{2} \cdot 2\sqrt{t} + C = \sqrt{t} + C. \]

### Step 5: Substitute back
Now, substituting back \( t = x + 1 \):

\[ I = \sqrt{x + 1} + C. \]

### Final Answer
Thus, the evaluated integral is:

\[ \int \frac{dx}{\sqrt{x+1} + \sqrt{x+1}} = \sqrt{x + 1} + C. \]