https://file.aiquickdraw.com/mathbot/file/1721643354107vy88dym0.png

To solve the integral given in the image, let's revisit the integral and evaluate it step-by-step.

The integral is:
\[ I = \int_0^\pi \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \]

To solve this, we will use the substitution \( u = \pi - x \). Let’s see how this helps us solve the integral:

1. **Substitution**: Let \( u = \pi - x \), then \( du = -dx \).

Rewriting the integral:
\[ I = \int_0^\pi \frac{(\pi - u) \sin^{2n} (\pi - u)}{\sin^{2n} (\pi - u) + \cos^{2n} (\pi - u)} (-du) \]

Since \(\sin(\pi - u) = \sin u\) and \(\cos(\pi - u) = -\cos u\), we get:
\[ I = \int_\pi^0 \frac{(\pi - u) \sin^{2n} u}{\sin^{2n} u + \cos^{2n} u} (-du) \]
\[ I = \int_0^\pi \frac{(\pi - u) \sin^{2n} u}{\sin^{2n} u + \cos^{2n} u} \, du \]

Now, we have two expressions for the integral \(I\):
\[ I = \int_0^\pi \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \]
\[ I = \int_0^\pi \frac{(\pi - x) \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \]

Adding these two expressions:
\[ 2I = \int_0^\pi \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx + \int_0^\pi \frac{(\pi - x) \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \]
\[ 2I = \int_0^\pi \frac{x \sin^{2n} x + (\pi - x) \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \]
\[ 2I = \int_0^\pi \frac{\pi \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \]

Therefore:
\[ 2I = \pi \int_0^\pi \frac{\sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \]

Since the integral on the right side is symmetric and periodic:
\[ \int_0^\pi \frac{\sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx = \frac{\pi}{2} \]

Thus:
\[ 2I = \pi \left( \frac{\pi}{2} \right) \]
\[ 2I = \frac{\pi^2}{2} \]
\[ I = \frac{\pi^2}{4} \]

So, the final answer is:
\[ I = \frac{\pi^2}{4} \]

If you have any further questions or need additional details, please let me know. 

Here are 5 more questions you can ask:
1. How to evaluate integrals involving trigonometric functions using substitutions?
2. How to determine the number of roots of a polynomial equation?
3. How to solve integrals using the method of symmetry?
4. What is the Intermediate Value Theorem and how is it applied?
5. How to find critical points of a function and their significance?

**Tip**: When solving integrals, always consider using symmetry and periodic properties of trigonometric functions to simplify the computation.

Learn how to solve the integral \( \int_0^\pi \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx \) using substitution and symmetry. Detailed step-by-step solution provided, covering integral calculus and trigonometric functions.

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