`int_(-pi//2)^(pi//2)sin^(10)x(6x^9-25x^7+4x^3-2x)dx` is equal to

Options

A. `pi`
B. `0`
C. 25
D. `pi/2`

Answer

To solve the integral

\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{10}(x) (6x^9 - 25x^7 + 4x^3 - 2x) \, dx, \]

we will use the property of definite integrals involving odd and even functions.

### Step 1: Identify the function inside the integral

Let

\[ f(x) = \sin^{10}(x) (6x^9 - 25x^7 + 4x^3 - 2x). \]

### Step 2: Check if \( f(x) \) is an odd function

To determine if \( f(x) \) is odd, we need to compute \( f(-x) \):

\[ f(-x) = \sin^{10}(-x) (6(-x)^9 - 25(-x)^7 + 4(-x)^3 - 2(-x)). \]

Using the properties of sine and powers of \( x \):

\[ \sin(-x) = -\sin(x) \quad \text{and} \quad (-x)^n = -x^n \text{ for odd } n, \]

we have:

\[ f(-x) = \sin^{10}(x) (-6x^9 + 25x^7 - 4x^3 + 2x). \]

### Step 3: Simplify \( f(-x) \)

This simplifies to:

\[ f(-x) = \sin^{10}(x) (-1) (6x^9 - 25x^7 + 4x^3 - 2x) = -f(x). \]

### Step 4: Conclude that \( f(x) \) is odd

Since \( f(-x) = -f(x) \), we conclude that \( f(x) \) is an odd function.

### Step 5: Evaluate the integral

The integral of an odd function over a symmetric interval around zero is zero:

\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x) \, dx = 0. \]

### Final Answer

Thus, the value of the integral is

\[ \boxed{0}. \]