Compute the integral
`int_(-7)^(7) (x^(4) sin x)/(x^(6) + 2) dx`

Answer

To compute the integral

\[ I = \int_{-7}^{7} \frac{x^4 \sin x}{x^6 + 2} \, dx, \]

we will use the property of definite integrals that relates the function evaluated at \(x\) and \(-x\).

### Step 1: Analyze the function

First, we need to check the behavior of the integrand under the transformation \(x \to -x\):

\[ f(-x) = \frac{(-x)^4 \sin(-x)}{(-x)^6 + 2}. \]

### Step 2: Simplify \(f(-x)\)

Now, simplify \(f(-x)\):

- The term \((-x)^4\) becomes \(x^4\).
- The term \(\sin(-x)\) becomes \(-\sin(x)\).
- The term \((-x)^6\) becomes \(x^6\).

Thus, we have:

\[ f(-x) = \frac{x^4 (-\sin x)}{x^6 + 2} = -\frac{x^4 \sin x}{x^6 + 2} = -f(x). \]

### Step 3: Use the property of definite integrals

Since \(f(-x) = -f(x)\), the function is odd. The integral of an odd function over a symmetric interval about zero is zero:

\[ I = \int_{-7}^{7} f(x) \, dx = 0. \]

### Conclusion

Therefore, the value of the integral is:

\[ \int_{-7}^{7} \frac{x^4 \sin x}{x^6 + 2} \, dx = 0. \]

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