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

Answer

To solve the integral

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

we will determine whether the integrand is an even function or an odd function.

### Step 1: Define the function
Let
\[ f(x) = \frac{\sin^5 x \cos^3 x}{x^4}. \]

### Step 2: Check if the function is even or odd
To check if \( f(x) \) is even or odd, we will compute \( f(-x) \):

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

Using the properties of sine and cosine:
- \(\sin(-x) = -\sin(x)\)
- \(\cos(-x) = \cos(x)\)

Substituting these into \( f(-x) \):

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

### Step 3: Simplify \( f(-x) \)
This simplifies to:

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

### Step 4: Conclusion about the function
Since \( f(-x) = -f(x) \), the function \( f(x) \) is an odd function.

### Step 5: Evaluate the integral
The integral of an odd function over a symmetric interval around zero is zero:

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

Thus, we conclude that:

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

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

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