If `int_(pi/6)^(pi/3) sqrt (1- sin 2x) dx = alpha + beta sqrt 2 + gamma sqrt 3`, where `alpha, beta` and `gamma` are rational numbers, then `3 alpha +4 beta-gamma` is equal to _______.

Answer

To solve the integral \( I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1 - \sin 2x} \, dx \), we will follow these steps:

### Step 1: Simplify the integrand
We know that:
\[ 1 - \sin 2x = 1 - 2 \sin x \cos x = \cos^2 x + \sin^2 x - 2 \sin x \cos x = (\cos x - \sin x)^2 \]
Thus, we can rewrite the integral as:
\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{(\cos x - \sin x)^2} \, dx \]
Since \( \cos x - \sin x \) is non-negative in the interval \( \left[\frac{\pi}{6}, \frac{\pi}{3}\right] \), we have:
\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} (\cos x - \sin x) \, dx \]

### Step 2: Compute the integral
Now we compute the integral:
\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} (\cos x - \sin x) \, dx \]
This can be split into two separate integrals:
\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos x \, dx - \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin x \, dx \]

### Step 3: Evaluate each integral
1. **Integral of \( \cos x \)**:
\[ \int \cos x \, dx = \sin x \]
Evaluating from \( \frac{\pi}{6} \) to \( \frac{\pi}{3} \):
\[ \left[ \sin x \right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = \sin\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{\sqrt{3} - 1}{2} \]

2. **Integral of \( \sin x \)**:
\[ \int \sin x \, dx = -\cos x \]
Evaluating from \( \frac{\pi}{6} \) to \( \frac{\pi}{3} \):
\[ \left[ -\cos x \right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = -\cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{\pi}{6}\right) = -\frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{\sqrt{3} - 1}{2} \]

### Step 4: Combine the results
Putting it all together:
\[ I = \left( \frac{\sqrt{3} - 1}{2} \right) - \left( \frac{\sqrt{3} - 1}{2} \right) = 0 \]

### Step 5: Express in the given form
Now, we need to express \( I \) in the form \( \alpha + \beta \sqrt{2} + \gamma \sqrt{3} \). Since \( I = 0 \), we have:
\[ \alpha = 0, \quad \beta = 0, \quad \gamma = 0 \]

### Step 6: Calculate \( 3\alpha + 4\beta - \gamma \)
Now we calculate:
\[ 3\alpha + 4\beta - \gamma = 3(0) + 4(0) - 0 = 0 \]

Thus, the final answer is:
\[ \boxed{0} \]