Question

Evaluate limₓtₒᵢₙftyfracint₀²x x eˣ²dxe⁴x²

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

Hint:

Look for substitution-friendly integrals before applying limits.

Answer 1

The Correct Option is D

\( \int x e^{x^2} \, dx = \dfrac{1}{2} e^{x^2} \)

\( \int_{0}^{x} x e^{x^2} \, dx = \dfrac{1}{2} \left( e^{x^2} - 1 \right) \) 

\( \Rightarrow \lim_{x \to \infty} \dfrac{\dfrac{1}{2}\left(e^{x^2} - 1\right)}{e^{x^2}} = \dfrac{1}{2} \)