Question

Consider the integrals $I_1={\int }_0^1{e}^{-x}{\cos }^{2}xdx,I_2={\int }_0^1{e}^{-x^2}{\cos }^{2}xdx,I_3={\int }_0^1{e}^{-x}dx$ and $I_4={\int }_0^1{e}^{-\left(1/2\right)x^2}dx$. The greatest of these integrals is

• A. $I_1$
• B. $I_2$
• C. $I_3$
• D. $I_4$

Answer 1

Answer: (D)

$I_4$

$I_1={\int }_0^4{e}^{-x}co{s}^{2}xdx$
$I_2={\int }_0^1{e}^{-x^2}co{s}^{2}xdx$
Both have $co{s}^{2}x$ so value get restricted more in (0, 1)
Now in (0, 1) $e^{-\frac{x^2}{2}}>e^{-x}$
$\therefore {\int }_0^1{e}^{-\frac{x^2}{2}}>{\int }_0^1{e}^{-x}$
$\therefore I_4>I_3>I_1>I_2$