$$  $$$$ \int_0^{\frac{3\pi}{4}}\!\sqrt{1-\sin2x}\,\mathrm{d}x $$​

It seems that your recent submission is the same integral as the one previously discussed. Here's a recap of the solution:

Given the integral:

\[
\int_0^{\frac{3\pi}{4}} \sqrt{1-\sin 2x} \, \mathrm{d}x
\]

### Steps to Solve:

1. **Simplify the integrand**: 
   The integral can be rewritten by using the trigonometric identity:
   \[
   1 - \sin 2x = \cos^2\left(x + \frac{\pi}{4}\right)
   \]
   Therefore, the integral becomes:
   \[
   \int_0^{\frac{3\pi}{4}} |\cos\left(x + \frac{\pi}{4}\right)| \, \mathrm{d}x
   \]
   The absolute value is necessary because $\cos(y)$ can be negative.

2. **Split the Integral**:
   The integral can be split based on where the cosine function changes its sign:
   \[
   \int_0^{\frac{\pi}{2}} \cos\left(x + \frac{\pi}{4}\right) \, \mathrm{d}x - \int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \cos\left(x + \frac{\pi}{4}\right) \, \mathrm{d}x
   \]

3. **Evaluate Each Integral**:
   - For the first part:
     \[
     \int_0^{\frac{\pi}{2}} \cos\left(x + \frac{\pi}{4}\right) \, \mathrm{d}x = \sin\left(x + \frac{\pi}{4}\right) \Big|_0^{\frac{\pi}{2}} = \sin\frac{3\pi}{4} - \sin\frac{\pi}{4} = 0
     \]
   - For the second part:
     \[
     -\int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \cos\left(x + \frac{\pi}{4}\right) \, \mathrm{d}x = \sin\left(x + \frac{\pi}{4}\right) \Big|_{\frac{\pi}{2}}^{\frac{3\pi}{4}} = \frac{\sqrt{2}}{2}
     \]

### Final Result:
The value of the integral is:
\[
\boxed{\frac{\sqrt{2}}{2}}
\]

Would you like to explore any step further or need any additional explanation?

---

1. What happens if the limits of the integral change?
2. How can different trigonometric identities simplify integrals?
3. What is the impact of the absolute value on the integrand?
4. Could this method be used to solve similar integrals?
5. Why is it important to consider the sign of the trigonometric function?

**Tip:** Always check for any potential sign changes when dealing with trigonometric integrals that involve absolute values to ensure the correct evaluation of the integral.

Learn how to evaluate the integral ∫₀^(3π/₄) √(1 - sin 2x) dx step-by-step. This problem involves trigonometric identities and the integral of cosine function, suitable for advanced college-level students.

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