Question

${\int }_0^{1}x{e}^{x^2}dx=\lambda {\int }_0^1{e}^{x^2}dx,$ then

• A. $\lambda =0$
  
• B. $\lambda \in (0,1)$
  
• C. $\lambda \in (-\infty ,0)$
  
  
• D. $\lambda \in (1,2)$

Answer 1

The correct option is B

$\lambda \in (0,1)$

Since $0<x<1$ and $e^{x^2}>0$
$\therefore 0<x{e}^{x^2}<e^{x^2}\therefore {\int }_0^{1}0dx<{\int }_0^{1}x{e}^{x^2}<{\int }_0^1{e}^{x^2}dx\Rightarrow 0<\lambda {\int }_0^1{e}^{x^2}dx<{\int }_0^1{e}^{x^2}dx\Rightarrow 0<\lambda <1\Rightarrow \lambda \in (0,1)$