Question

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

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

Answer 1

Answer: (B)

$\lambda ϵ=(0,1)$

For $x\in (0,1),x{e}^{x^2}<e^{x^2}$
$\Rightarrow {\int }_0^{1}x{e}^{x^2}<{\int }_0^1{e}^{x^2}dx$
Also $x{e}^{x^2}>0\forall x\in (0,1)$
$\Rightarrow {\int }_0^{1}x{e}^{x^2}>0$
Hence $\lambda \in (0,1)$