Question

The value of $\int_{-3}^{3} \sin^7 x \cos^{16} x \, dx$ is

• A. 1
• B. 2
• C. 0
• D. -1

Hint:

Check if the function is odd when you see limits like $\int_{-a}^{a}$; it often saves tedious calculation!

Answer 1

The Correct Option is C

Step 1: Concept
Integration of an odd function over a symmetric interval $[-a, a]$ is always zero.

Step 2: Meaning
A function $f(x)$ is odd if $f(-x) = -f(x)$.

Step 3: Analysis
Let $f(x) = \sin^7 x \cos^{16} x$. Then $f(-x) = [\sin(-x)]^7 [\cos(-x)]^{16} = [-\sin x]^7 [\cos x]^{16} = -\sin^7 x \cos^{16} x = -f(x)$.

Step 4: Conclusion
Since the function is odd and the limits are symmetric about the origin, the integral is 0. Final Answer: (C)