Question

Evaluate the following integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^4 + x^3 + x \right) \cos x \, dx \]

• A. 0
• B. \( \frac{2\pi}{3} \)
• C. \( \frac{\pi}{3} \)
• D. \( 1 \)

Hint:

When integrating odd functions over symmetric limits, the result is always zero. This property simplifies many integrals.

Answer 1

The Correct Option is A

We are asked to evaluate the integral: \[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^4 + x^3 + x \right) \cos x \, dx \] ### Step 1: Analyze the Symmetry of the Integrand The integrand \( f(x) = (x^4 + x^3 + x) \cos x \) is a product of two parts: 1. \( (x^4 + x^3 + x) \), which is an odd function, since \( x^4 \) is even, \( x^3 \) is odd, and \( x \) is odd. 2. \( \cos x \), which is an even function. When multiplying an odd function and an even function, the result is an odd function. Therefore, \( f(x) \) is odd. ### Step 2: Use the Property of Odd Functions For any odd function \( f(x) \), we know that: \[ \int_{-a}^{a} f(x) \, dx = 0 \] Since \( f(x) \) is odd and the limits of the integral are symmetric around zero (\( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \)), the integral evaluates to zero: \[ I = 0 \] Thus, the correct answer is: \[ \boxed{(A) 0} \]