Question

If \( f(x) \) is an odd function, then: \[ \int_{-\pi/2}^{\pi/2} f(x) \cos^3 x \, dx \, \text{equals:} \]

• A. \( 2 \int_0^{\pi/2} f(x) \cos^3 x \, dx \)
• B. \( 0 \)
• C. \( 2 \int_0^{\pi/2} f(x) \, dx \)
• D. \( 2 \int_0^{\pi/2} \cos^3 x \, dx \)

Hint:

The integral of an odd function over symmetric limits always results in zero.

Answer 1

The Correct Option is B

Step 1: Apply the property of odd functions. If \( f(x) \) is odd, then \( f(-x) = -f(x) \). Over symmetric limits, the integral of an odd function is zero: \[ \int_{-a}^{a} f(x) \, dx = 0 \] Step 2: Consider the given integral. \[ \int_{-\pi/2}^{\pi/2} f(x) \cos^3 x \, dx \] Since \( \cos^3 x \) is an even function and \( f(x) \) is odd, their product remains an odd function. Step 3: Conclude the result. Since the product is an odd function over symmetric limits: \[ \int_{-\pi/2}^{\pi/2} f(x) \cos^3 x \, dx = 0 \] Final Answer: \[ \boxed{0} \]