`int_(-pi//2) ^(pi//2)sin^(3) xdx=`

Options

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

Answer

To solve the definite integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^3 x \, dx \), we will follow these steps:

### Step 1: Define the Integral
Let \( A = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^3 x \, dx \).

### Step 2: Determine if the Function is Odd or Even
We need to check whether the function \( f(x) = \sin^3 x \) is odd or even.

- A function \( f(x) \) is odd if \( f(-x) = -f(x) \).
- Calculate \( f(-x) \):
\[ f(-x) = \sin^3(-x) = (-\sin x)^3 = -\sin^3 x = -f(x) \]
Thus, \( f(-x) = -f(x) \), which means \( f(x) = \sin^3 x \) is an odd function.

### Step 3: Apply the Property of Odd Functions
The property of definite integrals states that if \( f(x) \) is an odd function, then:
\[ \int_{-a}^{a} f(x) \, dx = 0 \]
for any \( a \).

### Step 4: Conclude the Integral
Since \( \sin^3 x \) is an odd function and we are integrating over the symmetric interval from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \):
\[ A = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^3 x \, dx = 0 \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^3 x \, dx = 0 \]