If $I_1=\int_0^1 e^{-x} \cos ^2 x d x ; I_2=\int_0^1 e^{-x^2} \cos ^2 x d x$ and $I_3=\int_0^1 e^{-x^3} d x$…
If $I_1=\int_0^1 e^{-x} \cos ^2 x d x ; I_2=\int_0^1 e^{-x^2} \cos ^2 x d x$ and $I_3=\int_0^1 e^{-x^3} d x$; then
-
$I_2>I_3>I_1$
-
$I_3>I_1>I_2$
-
$I_2>I_1>I_3$
-
$I_3>I_2>I_1$
Solution
Given:
$
\begin{aligned}
&I_1=\int_0^1 e^{-x} \cos ^2 x d x \\
&I_2=\int_0^1 e^{-x^2} \cos ^2 x d x \text { and } \\
&I_3=\int_0^1 e^{-x^3} d x \\
&\text { For } x \in(0,1) \\
&\Rightarrow x>x^2 \text { or }-x < -x^2 \\
&\text { and } x^2>x^3 \text { or }-x^2 < -x^3 \\
&\therefore e^{-x^2} < e^{-x^3} \text { and } e^{-x} < e^{-x^2} \\
&\Rightarrow e^{-x} < e^{-x^2} < e^{-x^3} \\
&\Rightarrow e^{-x^3}>e^{-x^2}>e^{-x} \\
&\Rightarrow I_3>I_2>I_1
\end{aligned}
$
Asked in: JEE Main 2018 (15 Apr Shift 2 Online)
Practice more Definite Integration questions on Aicharya