If I1-01e-xcos 2x dx-I2-01e-x2cos 2x dx andI3-01e-x3 dx - then -

As x∈ (0,1) So, x gt;x^2 gt;x^3 ⇒ x^3 x^2 x ⇒ e^ x^3 e^ x^2 e^ x ⇒ e^ x^3cos^2x e^ x^2cos^2x e^ x ∵cos^2x 1 ∀ x∈ (0,1) ⇒∫_0^1e^ x^3 dx∫_0^1e^ x^2c ...

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Byju's Answer

Standard XIII

Mathematics

Properties of Inequalities

If I1=∫01e-xc...

Question

If $I_{1} = 1 \int 0 e^{- x} {cos}^{2} x d x$ ,
$I_{2} = 1 \int 0 e^{- x^{2}} {cos}^{2} x d x$ and
$I_{3} = 1 \int 0 e^{- x^{3}} d x$ ; then :

I_{3} > I_{2} > I_{1}

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I_{2} > I_{1} > I_{3}

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I_{2} > I_{3} > I_{1}

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I_{3} > I_{1} > I_{2}

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Solution

The correct option is A $I_{3} > I_{2} > I_{1}$
As $x \in (0, 1)$
So, $x > x^{2} > x^{3}$
$\Rightarrow - x^{3} < - x^{2} < - x$
$\Rightarrow e^{- x^{3}} > e^{- x^{2}} > e^{- x}$
$\Rightarrow e^{- x^{3}} > {cos}^{2} x e^{- x^{2}} > {cos}^{2} x e^{- x}$
$∵ {cos}^{2} x < 1 \forall x \in (0, 1)$
$\Rightarrow 1 \int 0 e^{- x^{3}} d x > 1 \int 0 e^{- x^{2}} {cos}^{2} x d x > 1 \int 0 e^{- x} {cos}^{2} x d x$
$∴ I_{3} > I_{2} > I_{1}$

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Similar questions

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If I1=1∫0e−xcos2x dx, I2=1∫0e−x2cos2x dx and I3=1∫0e−x3 dx ; then :

The correct option is A I3>I2>I1As x∈(0,1) So, x>x2>x3 ⇒−x3<−x2<−x ⇒e−x3>e−x2>e−x ⇒e−x3>cos2x e−x2>cos2x e−x ∵cos2x<1 ∀ x∈(0,1) ⇒1∫0e−x3 dx>1∫0e−x2cos2x dx>1∫0e−xcos2x dx ∴I3>I2>I1

If $I_{1} = 1 \int 0 e^{- x} {cos}^{2} x d x$ ,
$I_{2} = 1 \int 0 e^{- x^{2}} {cos}^{2} x d x$ and
$I_{3} = 1 \int 0 e^{- x^{3}} d x$ ; then :