Question

The value of integral \( \int_{-2}^{2} x e^{-2x^2} dx \) is

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

Hint:

For integrals of odd functions over symmetric limits, the result is always zero.

Answer 1

The Correct Option is A

Step 1: Identifying the symmetry.
The given integral involves an odd function (\( x e^{-2x^2} \)) integrated over a symmetric interval from \( -2 \) to \( 2 \). The integral of an odd function over a symmetric interval is always 0.

Step 2: Analyzing the options.
(A) 0: Correct — The integral of an odd function over a symmetric interval is zero.
(B) \( \frac{1}{2} \): Incorrect — This is not the correct value of the integral.
(C) 1: Incorrect — The result is not 1.
(D) 2: Incorrect — This is not the correct value of the integral.

Step 3: Conclusion.
The correct answer is (A) 0 because the integrand is an odd function, and its integral over a symmetric interval is zero.