`int_(0)^(1)(1)/(sqrt(x+3)+sqrt(x+2))dx=`

Options

A. `(4)/(3)(sqrt(2)-3sqrt(3)+4)`
B. `(2)/(3)(8-6sqrt(3))`
C. `2+sqrt(3)`
D. `(4)/(3)(sqrt(2)+3sqrt(3)-4)`

Answer

To solve the integral \( I = \int_{0}^{1} \frac{1}{\sqrt{x+3} + \sqrt{x+2}} \, dx \), we will follow a series of steps to simplify and evaluate it.

### Step-by-Step Solution:

1. **Set up the integral**:
\[ I = \int_{0}^{1} \frac{1}{\sqrt{x+3} + \sqrt{x+2}} \, dx \]

2. **Rationalize the denominator**:
Multiply the numerator and denominator by \( \sqrt{x+3} - \sqrt{x+2} \):
\[ I = \int_{0}^{1} \frac{\sqrt{x+3} - \sqrt{x+2}}{(\sqrt{x+3} + \sqrt{x+2})(\sqrt{x+3} - \sqrt{x+2})} \, dx \]

3. **Simplify the denominator**:
The denominator simplifies as follows:
\[ (\sqrt{x+3})^2 - (\sqrt{x+2})^2 = (x+3) - (x+2) = 1 \]
Thus, the integral becomes:
\[ I = \int_{0}^{1} (\sqrt{x+3} - \sqrt{x+2}) \, dx \]

4. **Separate the integral**:
\[ I = \int_{0}^{1} \sqrt{x+3} \, dx - \int_{0}^{1} \sqrt{x+2} \, dx \]

5. **Evaluate the first integral**:
Let \( t = x + 3 \) then \( dt = dx \). The limits change from \( x=0 \) to \( t=3 \) and from \( x=1 \) to \( t=4 \):
\[ \int_{0}^{1} \sqrt{x+3} \, dx = \int_{3}^{4} \sqrt{t} \, dt = \left[ \frac{2}{3} t^{3/2} \right]_{3}^{4} \]
Evaluating this gives:
\[ = \frac{2}{3} \left( 4^{3/2} - 3^{3/2} \right) = \frac{2}{3} \left( 8 - 3\sqrt{3} \right) \]

6. **Evaluate the second integral**:
Let \( z = x + 2 \) then \( dz = dx \). The limits change from \( x=0 \) to \( z=2 \) and from \( x=1 \) to \( z=3 \):
\[ \int_{0}^{1} \sqrt{x+2} \, dx = \int_{2}^{3} \sqrt{z} \, dz = \left[ \frac{2}{3} z^{3/2} \right]_{2}^{3} \]
Evaluating this gives:
\[ = \frac{2}{3} \left( 3^{3/2} - 2^{3/2} \right) = \frac{2}{3} \left( 3\sqrt{3} - 2\sqrt{2} \right) \]

7. **Combine the results**:
\[ I = \frac{2}{3} \left( 8 - 3\sqrt{3} \right) - \frac{2}{3} \left( 3\sqrt{3} - 2\sqrt{2} \right) \]
Simplifying this:
\[ I = \frac{2}{3} \left( 8 - 3\sqrt{3} - 3\sqrt{3} + 2\sqrt{2} \right) = \frac{2}{3} \left( 8 - 6\sqrt{3} + 2\sqrt{2} \right) \]

8. **Final result**:
\[ I = \frac{2}{3} \left( 8 + 2\sqrt{2} - 6\sqrt{3} \right) \]