`int_0^1 1/(sqrt(x+3)-sqrt(x+1)) dx`

Answer

To solve the integral \( I = \int_0^1 \frac{1}{\sqrt{x+3} - \sqrt{x+1}} \, dx \), we will follow these steps:

### Step 1: Rationalize the Denominator
We start by multiplying the numerator and denominator by the conjugate of the denominator:

\[ I = \int_0^1 \frac{1}{\sqrt{x+3} - \sqrt{x+1}} \cdot \frac{\sqrt{x+3} + \sqrt{x+1}}{\sqrt{x+3} + \sqrt{x+1}} \, dx \]

This gives us:

\[ I = \int_0^1 \frac{\sqrt{x+3} + \sqrt{x+1}}{(\sqrt{x+3})^2 - (\sqrt{x+1})^2} \, dx \]

### Step 2: Simplify the Denominator
Now we simplify the denominator:

\[ (\sqrt{x+3})^2 - (\sqrt{x+1})^2 = (x+3) - (x+1) = 2 \]

Thus, we can rewrite the integral as:

\[ I = \int_0^1 \frac{\sqrt{x+3} + \sqrt{x+1}}{2} \, dx \]

### Step 3: Factor Out the Constant
We can factor out the constant \( \frac{1}{2} \):

\[ I = \frac{1}{2} \int_0^1 (\sqrt{x+3} + \sqrt{x+1}) \, dx \]

### Step 4: Split the Integral
Now we can split the integral into two parts:

\[ I = \frac{1}{2} \left( \int_0^1 \sqrt{x+3} \, dx + \int_0^1 \sqrt{x+1} \, dx \right) \]

### Step 5: Evaluate Each Integral
We will evaluate each integral separately.

#### Integral 1: \( \int_0^1 \sqrt{x+3} \, dx \)
Let \( u = x + 3 \), then \( du = dx \) and the limits change from \( x=0 \) to \( u=3 \) and from \( x=1 \) to \( u=4 \):

\[ \int_0^1 \sqrt{x+3} \, dx = \int_3^4 \sqrt{u} \, du = \left[ \frac{2}{3} u^{3/2} \right]_3^4 = \frac{2}{3} (4^{3/2} - 3^{3/2}) = \frac{2}{3} (8 - 3\sqrt{3}) \]

#### Integral 2: \( \int_0^1 \sqrt{x+1} \, dx \)
Let \( v = x + 1 \), then \( dv = dx \) and the limits change from \( x=0 \) to \( v=1 \) and from \( x=1 \) to \( v=2 \):

\[ \int_0^1 \sqrt{x+1} \, dx = \int_1^2 \sqrt{v} \, dv = \left[ \frac{2}{3} v^{3/2} \right]_1^2 = \frac{2}{3} (2^{3/2} - 1^{3/2}) = \frac{2}{3} (2\sqrt{2} - 1) \]

### Step 6: Combine the Results
Now we combine the results of the two integrals:

\[ I = \frac{1}{2} \left( \frac{2}{3} (8 - 3\sqrt{3}) + \frac{2}{3} (2\sqrt{2} - 1) \right) \]

\[ I = \frac{1}{3} (8 - 3\sqrt{3} + 2\sqrt{2} - 1) = \frac{1}{3} (7 - 3\sqrt{3} + 2\sqrt{2}) \]

### Final Result
Thus, the value of the integral is:

\[ I = \frac{7 - 3\sqrt{3} + 2\sqrt{2}}{3} \]