Question

The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7}x \, dx \] is

• A. 1
• B. 0
• C. $-1$
• D. 7

Hint:

If $f(x)$ is odd, then $\int_{-a}^{a} f(x)\,dx = 0$.

Answer 1

The Correct Option is B

Step 1: Observe the nature of the function.
The integrand is \[ f(x) = \sin^{7}x \] We know that the sine function is an odd function.
An odd function satisfies the property \[ f(-x) = -f(x) \]
Step 2: Check whether the integrand is odd.
Since $\sin x$ is odd, \[ \sin(-x) = -\sin x \] Thus, \[ \sin^{7}(-x) = (-\sin x)^7 \] \[ = -\sin^{7}x \] Hence $\sin^{7}x$ is also an odd function.

Step 3: Use the property of definite integrals of odd functions.
If a function is odd and the limits of integration are symmetric about zero, \[ \int_{-a}^{a} f(x)\,dx = 0 \] Here the limits are \[ -\frac{\pi}{2} \quad \text{to} \quad \frac{\pi}{2} \] which are symmetric about zero.

Step 4: Apply the property.
Therefore, \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7}x \, dx = 0 \]
Step 5: Conclusion.
Thus the value of the definite integral is zero.
Final Answer: $\boxed{0}$