The value of `int_(-pi/8)^(pi/8) x^(10) sin^(9) x dx` is equal to

Options

A. `0`
B. `1`
C. `pi/4`
D. `(pi/8)^(10)`

Answer

To solve the integral \( I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} x^{10} \sin^9 x \, dx \), we can use the property of definite integrals concerning odd and even functions.

### Step-by-Step Solution:

1. **Identify the Function**:
Let \( f(x) = x^{10} \sin^9 x \). We need to determine whether \( f(x) \) is an odd or even function.

2. **Check for Evenness or Oddness**:
- An even function satisfies \( f(-x) = f(x) \).
- An odd function satisfies \( f(-x) = -f(x) \).

Now, calculate \( f(-x) \):
\[ f(-x) = (-x)^{10} \sin^9(-x) \]
Since \( (-x)^{10} = x^{10} \) (even power) and \( \sin(-x) = -\sin(x) \) (odd function), we have:
\[ f(-x) = x^{10} (-\sin^9 x) = -x^{10} \sin^9 x = -f(x) \]
This shows that \( f(x) \) is an odd function.

3. **Apply the Property of Definite Integrals**:
The integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd.} \]
Therefore,
\[ I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} f(x) \, dx = 0. \]

4. **Conclusion**:
The value of the integral \( I \) is:
\[ \boxed{0}. \]