`int_(3)^(5) (dx)/(sqrt(x+4)+sqrt(x-2))=`

Options

A. `1/9 (28-3sqrt(3) - 7 sqrt(7))`
B. `1/9 (28-3sqrt(3) + 7 sqrt(7))`
C. `1/9 (28+3sqrt(3) -7 sqrt(7))`
D. `1/9 (28+3sqrt(3) +7 sqrt(7))`

Answer

To solve the definite integral

\[ \int_{3}^{5} \frac{dx}{\sqrt{x+4} + \sqrt{x-2}}, \]

we will follow these steps:

### Step 1: Rationalize the Denominator
We start by multiplying the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{x+4} - \sqrt{x-2}\):

\[ \int_{3}^{5} \frac{dx}{\sqrt{x+4} + \sqrt{x-2}} \cdot \frac{\sqrt{x+4} - \sqrt{x-2}}{\sqrt{x+4} - \sqrt{x-2}}. \]

This gives us:

\[ \int_{3}^{5} \frac{\sqrt{x+4} - \sqrt{x-2}}{(\sqrt{x+4})^2 - (\sqrt{x-2})^2} \, dx. \]

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

\[ (\sqrt{x+4})^2 - (\sqrt{x-2})^2 = (x+4) - (x-2) = x + 4 - x + 2 = 6. \]

So, the integral simplifies to:

\[ \int_{3}^{5} \frac{\sqrt{x+4} - \sqrt{x-2}}{6} \, dx = \frac{1}{6} \int_{3}^{5} (\sqrt{x+4} - \sqrt{x-2}) \, dx. \]

### Step 3: Split the Integral
We can split the integral into two parts:

\[ \frac{1}{6} \left( \int_{3}^{5} \sqrt{x+4} \, dx - \int_{3}^{5} \sqrt{x-2} \, dx \right). \]

### Step 4: Compute Each Integral
Now we compute each integral separately.

1. **Integral of \(\sqrt{x+4}\)**:

Using the formula \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\):

\[ \int \sqrt{x+4} \, dx = \int (x+4)^{1/2} \, dx = \frac{2}{3}(x+4)^{3/2}. \]

Evaluating from 3 to 5:

\[ \left[ \frac{2}{3}(x+4)^{3/2} \right]_{3}^{5} = \frac{2}{3} \left[ (5+4)^{3/2} - (3+4)^{3/2} \right] = \frac{2}{3} \left[ 9^{3/2} - 7^{3/2} \right]. \]

Calculating \(9^{3/2} = 27\) and \(7^{3/2} = 7\sqrt{7}\).

2. **Integral of \(\sqrt{x-2}\)**:

Similarly,

\[ \int \sqrt{x-2} \, dx = \int (x-2)^{1/2} \, dx = \frac{2}{3}(x-2)^{3/2}. \]

Evaluating from 3 to 5:

\[ \left[ \frac{2}{3}(x-2)^{3/2} \right]_{3}^{5} = \frac{2}{3} \left[ (5-2)^{3/2} - (3-2)^{3/2} \right] = \frac{2}{3} \left[ 3^{3/2} - 1^{3/2} \right]. \]

Calculating \(3^{3/2} = 3\sqrt{3}\) and \(1^{3/2} = 1\).

### Step 5: Combine Results
Now we combine the results:

\[ \frac{1}{6} \left( \frac{2}{3} \left[ 27 - 7\sqrt{7} \right] - \frac{2}{3} \left[ 3\sqrt{3} - 1 \right] \right). \]

This simplifies to:

\[ \frac{1}{9} \left[ 27 - 7\sqrt{7} - 3\sqrt{3} + 1 \right] = \frac{1}{9} \left[ 28 - 7\sqrt{7} - 3\sqrt{3} \right]. \]

### Final Answer
Thus, the final answer is:

\[ \frac{28 - 7\sqrt{7} - 3\sqrt{3}}{9}. \]