`int_(-pi//2)^(pi//2)sin^(7)xdx`

Answer

To solve the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x \, dx\), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = \sin^7 x \).

### Step 2: Check if the function is odd or even
To determine if \( f(x) \) is odd or even, we need to evaluate \( f(-x) \):
\[ f(-x) = \sin^7(-x) = (-\sin x)^7 = -\sin^7 x = -f(x) \]
Since \( f(-x) = -f(x) \), we conclude that \( f(x) \) is an odd function.

### Step 3: Apply the property of definite integrals for odd functions
We know that the integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd} \]
In our case, since \( f(x) \) is odd and the interval is \([- \frac{\pi}{2}, \frac{\pi}{2}]\), we can conclude:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x \, dx = 0 \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x \, dx = 0 \]

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