Question

Let $I_1={\int }_0^1{e}^{-x^2}dx,I_2={\int }_0^1{e}^{-x}{\cos }^{2}xdx$ and $I_3={\int }_0^1{e}^{-x^2}{\cos }^{2}xdx.$ Then

• A. $I_1<I_2<I_3$
• B. $I_3<I_2<I_1$
• C. $I_2<I_1<I_3$
• D. $I_2<I_3<I_1$

Answer 1

The correct option is D $I_2<I_3<I_1$

When $0<x<1$
$x^2<x\Rightarrow e^{x^2}<e^x\Rightarrow e^{-x^2}>e^{-x}$
Multiply both sides by ${\cos }^{2}x$ we get
$e^{{-x}^2}{\cos }^{2}x>e^{-x}{\cos }^{2}x\Rightarrow {\int }_0^1{e}^{{-x}^2}{\cos }^{2}xdx>{\int }_0^1{e}^{-x}{\cos }^{2}xdx\Rightarrow I_3>I_2$
Similarly since $0<{\cos }^{2}x<1$
$\frac{1}{e^{x^2}}>\frac{{\cos }^{2}x}{e^{x^2}}\Rightarrow I_1>I_3\therefore I_2<I_3<I_1$