MCQ
If ${I_1} = \int\limits_0^1 {{e^{ - x}}} {\cos ^2}\,x\,dx\,;\,{I_2} = \int\limits_0^1 {{e^{ - {x^2}}}} {\cos ^2}\,x\,dx$ and $\,{I_3} = \int\limits_0^1 {{e^{ - {x^3}}}} dx$ ; then
- A$I_2>I_3>I_1$
- B$I_3>I_1>I_2$
- C$I_2>I_1>I_3$
- ✓$I_3>I_2>I_1$
$I_{1}=\int_{0}^{1} e^{-x} \cos ^{2} x d x$
$I_{2}=\int_{0}^{1} e^{-x^{2}} \cos ^{2} x d x$ and
$I_{3}=\int_{0}^{1} e^{-x^{3}} d x$
For $x \in(0,1)$
$\Rightarrow x>x^{2}$ or $-x<-x^{2}$
and $x^{2}>x^{3}$ or $-x^{2}<-x^{3}$
$\therefore {e^{ - {x^2}}} < {e^{ - {x^3}}}$ and ${e^{ - x}} < {e^{ - {x^2}}}$
$ \Rightarrow {e^{ - x}} < {e^{ - {x^2}}} < {e^{ - x}}$
$\Rightarrow e^{-x^{3}}>e^{-x^{2}}>e^{-x}$
$\Rightarrow I_{3}>I_{2}>I_{1}$
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