What is `int_(-pi/6)^(pi/6) (sin^(5)x cos^(3)x)/(x^(4)) dx` equal to ?

Options

A. `pi/2`
B. `pi/4`
C. `pi/8`
D. `0`

Explanation

To solve the integral

\[ I = \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\sin^5 x \cos^3 x}{x^4} \, dx, \]

we will determine if the integrand is an odd function.

### Step 1: Define the function
Let

\[ f(x) = \frac{\sin^5 x \cos^3 x}{x^4}. \]

### Step 2: Check for odd function
To check if \( f(x) \) is odd, we need to compute \( f(-x) \):

\[ f(-x) = \frac{\sin^5(-x) \cos^3(-x)}{(-x)^4}. \]

Using the properties of sine and cosine:

- \( \sin(-x) = -\sin(x) \)
- \( \cos(-x) = \cos(x) \)

we have:

\[ f(-x) = \frac{(-\sin x)^5 (\cos x)^3}{x^4} = \frac{-\sin^5 x \cos^3 x}{x^4} = -f(x). \]

### Step 3: Conclusion about the function
Since \( f(-x) = -f(x) \), we conclude that \( f(x) \) is an odd function.

### Step 4: Evaluate the integral
For any odd function \( f(x) \), the integral over a symmetric interval around zero is zero:

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

Thus, we have:

\[ I = \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} f(x) \, dx = 0. \]

### Final Answer
Therefore,

\[ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\sin^5 x \cos^3 x}{x^4} \, dx = 0. \]

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