Question

If \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \sin^4 x \, dx = k \), then \( k \) is ____________.

• A. \( 1 \)
• B. \( 2 \)
• C. \( 4 \)
• D. \( 0 \)

Hint:

The integral of an odd function over symmetric limits always equals zero: \[ \int_{-a}^{a} f(x) \, dx = 0 \]

Answer 1

The Correct Option is D

Step 1: Analyzing the Function's Parity 
The function inside the integral is: \[ f(x) = x^3 \sin^4 x \] Since \( x^3 \) is odd and \( \sin^4 x \) is even, their product \( x^3 \sin^4 x \) is an odd function. 
Step 2: Integrating an Odd Function 
For any odd function \( f(x) \), we know that: \[ \int_{-a}^{a} f(x) \, dx = 0 \] Given that our limits are symmetric about zero, we can directly conclude: \[ k = 0 \]