Question

If \( \lim_{t \to \infty} \int_0^t e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \), then \[ \lim_{t \to \infty} \int_0^t x^2 e^{-x^2} dx = \]

• A. \( \frac{\sqrt{\pi}}{4} \)
• B. \( \frac{\sqrt{\pi}}{2} \)
• C. \( \sqrt{2\pi} \)
• D. \( 2\sqrt{\pi} \)

Hint:

When dealing with Gaussian integrals, recognize the relationship between the functions and use integration by parts or substitutions effectively.

Answer 1

The Correct Option is A

Step 1: Recognizing the relationship between integrals.
The function \( x^2 e^{-x^2} \) is the derivative of \( -\frac{1}{2} e^{-x^2} \). Thus, we use integration by parts to evaluate the integral: \[ \int_0^t x^2 e^{-x^2} dx = \frac{1}{2} \left( - e^{-t^2} \right). \]
Step 2: Taking the limit.
Taking the limit as \( t \to \infty \), we get: \[ \lim_{t \to \infty} \int_0^t x^2 e^{-x^2} dx = \frac{\sqrt{\pi}}{4}. \]
Step 3: Conclusion.
Thus, the correct answer is (A).