Question

Evaluate: \(\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x \, dx = \)

• A. \(-1\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

For integrals of odd functions over symmetric limits \([-a, a]\), the integral is always zero: \[ \int_{-a}^a f(x) \, dx = 0 \quad \text{if } f(-x) = -f(x) \]

Answer 1

The Correct Option is B

Step 1: Note that \(\sin^7 x\) is an odd function because \(\sin x\) is odd and raising to an odd power preserves oddness: \[ \sin^7 (-x) = -\sin^7 x \] Step 2: The integral of an odd function over symmetric limits \([-a, a]\) is zero: \[ \int_{-a}^a f(x) \, dx = 0 \quad \text{if } f \text{ is odd} \] Step 3: Therefore, \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x \, dx = 0 \]