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To evaluate the integral 

\[
\int_0^4 \frac{\sin(x^2) \, dx}{\sin(x - 4)^2 + \sin(x^2)},
\]

let’s analyze the symmetry of the integrand.

### Step-by-Step Solution:
1. **Define the Integral:**
   Let \( I = \int_0^4 \frac{\sin(x^2)}{\sin((x-4)^2) + \sin(x^2)} \, dx \).

2. **Use Symmetry:**  
   To evaluate this, consider the substitution \( x \to 4 - x \). With this substitution:
   \[
   dx \to -dx.
   \]
   
   So,
   \[
   I = \int_0^4 \frac{\sin((4 - x)^2)}{\sin(x^2) + \sin((4 - x)^2)} \, dx.
   \]

3. **Add the Original and Transformed Integrals:**
   Now, let’s add the original integral and the transformed integral:
   
   \[
   I = \int_0^4 \frac{\sin(x^2)}{\sin((x-4)^2) + \sin(x^2)} \, dx + \int_0^4 \frac{\sin((4 - x)^2)}{\sin((4 - x)^2) + \sin(x^2)} \, dx.
   \]

   This gives:
   \[
   2I = \int_0^4 \frac{\sin(x^2) + \sin((4 - x)^2)}{\sin(x^2) + \sin((4 - x)^2)} \, dx.
   \]

4. **Simplify the Expression:**
   Since \(\sin(x^2) + \sin((4 - x)^2)\) appears in both the numerator and denominator, the fraction simplifies to \(1\):
   
   \[
   2I = \int_0^4 1 \, dx.
   \]

5. **Evaluate the Integral:**
   Now we have:
   \[
   2I = \int_0^4 1 \, dx = 4.
   \]
   
   Therefore,
   \[
   I = \frac{4}{2} = 2.
   \]

### Final Answer:
\[
\int_0^4 \frac{\sin(x^2)}{\sin((x-4)^2) + \sin(x^2)} \, dx = 2.
\]

Would you like a more detailed explanation, or do you have any questions? 

---

Here are some related questions for further practice:
1. How would the evaluation change if the limits of integration were different?
2. What if the integrand involved \(\cos(x^2)\) instead of \(\sin(x^2)\)?
3. How can you apply similar symmetry arguments to integrals involving trigonometric functions?
4. What are the properties of definite integrals that allow for substitutions like \( x \to 4 - x \)?
5. How would you evaluate this integral if the expression in the denominator were different, such as \(\sin(x - 4)^2 + \cos(x^2)\)?

**Tip:** Symmetry is a powerful tool in calculus, especially for evaluating definite integrals. Look for substitutions that simplify the integrand or result in terms that can be combined.

Evaluate the integral \( \int_0^4 \frac{\sin(x^2) dx}{\sin((x-4)^2) + \sin(x^2)} \)

Learn how to evaluate the integral \( \int_0^4 \frac{\sin(x^2)}{\sin((x-4)^2) + \sin(x^2)} \, dx \) using symmetry and substitution. A detailed step-by-step solution suitable for advanced calculus students.

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