Question

$$ \int_{-\pi/4}^{\pi/4} \sin^{103} x \cos^{101} x \, dx = ? $$

• A. $\left( \frac{\pi}{4} \right)^{103}$
• B. $\left( \frac{\pi}{4} \right)^{101}$
• C. $2$
• D. 0

Hint:

Whenever you see a symmetric interval like $[-a, a]$, always check if the function is odd. It often turns complex-looking trigonometric powers into a simple answer of zero.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem uses the property of definite integrals for even and odd functions. If a function $f(x)$ is odd, meaning $f(-x) = -f(x)$, then the integral over a symmetric interval $[-a, a]$ is zero.

Step 2: Key Formula or Approach:
Identify if the integrand $f(x) = \sin^{103} x \cos^{101} x$ is even or odd.

Step 3: Detailed Explanation:
1. Let $f(x) = \sin^{103} x \cos^{101} x$. 2. Check $f(-x)$: \[ f(-x) = \sin^{103}(-x) \cos^{101}(-x) \] 3. Since $\sin(-x) = -\sin x$ and $\cos(-x) = \cos x$: \[ f(-x) = (-\sin x)^{103} (\cos x)^{101} \] 4. Because 103 is an odd exponent: \[ f(-x) = -\sin^{103} x \cos^{101} x = -f(x) \] 5. Since $f(x)$ is an odd function, the integral over the symmetric interval $[-\pi/4, \pi/4]$ must be zero.

Step 4: Final Answer
The value of the integral is 0.