Question

The value of \( \int_{-1}^1 \sin^5 x \cos^4 x \, dx \) is:

• A. \( \pi \)
• B. \( \frac{\pi}{2} \)
• C. \( 0 \)
• D. \( -\pi \)

Hint:

When integrating an odd function over a symmetric interval, the integral always equals zero.

Answer 1

The Correct Option is C

We are given the integral: \[ \int_{-1}^1 \sin^5 x \cos^4 x \, dx \] We know that: \[ f(x) = \sin^5 x \cos^4 x \] Now, evaluate \( f(-x) \): \[ f(-x) = -\sin^5 x \cos^4 x = -f(x) \] Since \( f(-x) = -f(x) \), this is an odd function. The integral of an odd function over a symmetric interval is always zero: \[ \int_{-1}^1 f(x) \, dx = 0 \] Thus, the correct answer is \( 0 \).