Compute the integral `int_(-pi)^(pi)(sin 2 x)/( cos^(4) x + sin^(4) x)` dx

Answer

To compute the integral

\[ I = \int_{-\pi}^{\pi} \frac{\sin(2x)}{\cos^4(x) + \sin^4(x)} \, dx, \]

we can use the property of definite integrals that states:

\[ \int_{-a}^{a} f(x) \, dx = \int_{0}^{a} (f(x) + f(-x)) \, dx. \]

### Step 1: Find \( f(-x) \)

First, we need to find \( f(-x) \):

\[ f(-x) = \frac{\sin(2(-x))}{\cos^4(-x) + \sin^4(-x)}. \]

Using the properties of sine and cosine:

\[ \sin(-2x) = -\sin(2x) \quad \text{and} \quad \cos(-x) = \cos(x), \]

we can rewrite \( f(-x) \):

\[ f(-x) = \frac{-\sin(2x)}{\cos^4(x) + \sin^4(x)}. \]

### Step 2: Combine \( f(x) \) and \( f(-x) \)

Now, we can combine \( f(x) \) and \( f(-x) \):

\[ f(x) + f(-x) = \frac{\sin(2x)}{\cos^4(x) + \sin^4(x)} + \frac{-\sin(2x)}{\cos^4(x) + \sin^4(x)} = 0. \]

### Step 3: Evaluate the integral

Since \( f(x) + f(-x) = 0 \), we have:

\[ I = \int_{-\pi}^{\pi} f(x) \, dx = \int_{0}^{\pi} (f(x) + f(-x)) \, dx = \int_{0}^{\pi} 0 \, dx = 0. \]

Thus, the value of the integral is:

\[ \boxed{0}. \]