If `int_(0)^(1)(1)/(sqrt(3+x)+sqrt(1+x))dx=a+b sqrt(2)+c sqrt(3)` ,where `a,b,c` are rational numbers, then `2a+3b-4c`, is equal to:

Options

A. 7
B. 10
C. 8
D. 4

Answer

To solve the integral

\[ I = \int_{0}^{1} \frac{1}{\sqrt{3+x} + \sqrt{1+x}} \, dx, \]

we will use the technique of rationalization.

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

\[ I = \int_{0}^{1} \frac{\sqrt{3+x} - \sqrt{1+x}}{(\sqrt{3+x} + \sqrt{1+x})(\sqrt{3+x} - \sqrt{1+x})} \, dx. \]

The denominator simplifies as follows:

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

Thus, we have:

\[ I = \int_{0}^{1} \frac{\sqrt{3+x} - \sqrt{1+x}}{2} \, dx = \frac{1}{2} \int_{0}^{1} (\sqrt{3+x} - \sqrt{1+x}) \, dx. \]

### Step 2: Split the Integral
Now we can split the integral:

\[ I = \frac{1}{2} \left( \int_{0}^{1} \sqrt{3+x} \, dx - \int_{0}^{1} \sqrt{1+x} \, dx \right). \]

### Step 3: Evaluate Each Integral
1. **Integral of \(\sqrt{3+x}\)**:

Using the substitution \(u = 3+x\), \(du = dx\), the limits change from \(x=0\) to \(u=3\) and from \(x=1\) to \(u=4\):

\[ \int \sqrt{3+x} \, dx = \int \sqrt{u} \, du = \frac{2}{3} u^{3/2} + C. \]

Evaluating from 3 to 4:

\[ \left[ \frac{2}{3} u^{3/2} \right]_{3}^{4} = \frac{2}{3} (4^{3/2} - 3^{3/2}) = \frac{2}{3} (8 - 3\sqrt{3}). \]

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

Using the substitution \(v = 1+x\), \(dv = dx\), the limits change from \(x=0\) to \(v=1\) and from \(x=1\) to \(v=2\):

\[ \int \sqrt{1+x} \, dx = \int \sqrt{v} \, dv = \frac{2}{3} v^{3/2} + C. \]

Evaluating from 1 to 2:

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

### Step 4: Combine the Results
Now substituting back into the expression for \(I\):

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

Simplifying:

\[ I = \frac{1}{2} \cdot \frac{2}{3} \left( 8 - 3\sqrt{3} - 2\sqrt{2} + 1 \right) = \frac{1}{3} (9 - 3\sqrt{3} - 2\sqrt{2}). \]

### Step 5: Final Expression
Thus,

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

### Step 6: Identify \(a\), \(b\), and \(c\)
From the expression \(I = a + b\sqrt{2} + c\sqrt{3}\):

- \(a = 3\)
- \(b = -\frac{2}{3}\)
- \(c = -1\)

### Step 7: Calculate \(2a + 3b - 4c\)
Now we compute:

\[ 2a + 3b - 4c = 2(3) + 3\left(-\frac{2}{3}\right) - 4(-1). \]

Calculating each term:

\[ = 6 - 2 + 4 = 8. \]

Thus, the final answer is:

\[ \boxed{8}. \]