If I1-0- - 2cos sin x d x - I2-0- - 2sin cos x d x and I3-0- - 2cos x d x- then

For x∈(0, π2), we know that ⇒cos xsin (cos x) [∵ f(x)=cos x sin(cos x) is decreasing in given interval] ⇒ nbsp; I_3 gt;I_2 nbsp; Also, xsin x ⇒cos (sin x ...

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Byju's Answer

Standard XII

Mathematics

Geometric Interpretation of Def.Int as Limit of Sum

If I1=∫0π / 2...

Question

If $I_{1} = π / 2 \int 0 cos (sin x) d x, I_{2} = π / 2 \int 0 sin (cos x) d x$ and $I_{3} = π / 2 \int 0 cos x d x$ , then

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

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

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

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

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Solution

The correct option is A $I_{1} > I_{3} > I_{2}$
For $x \in (0, \frac{π}{2})$ ,
we know that
$\Rightarrow cos x > sin (cos x) [∵ f (x) = cos x - sin (cos x) is decreasing in given interval] \Rightarrow I_{3} > I_{2}$
Also,
$x > sin x \Rightarrow cos (sin x) > cos x [∵ cos x is decreasing function] \Rightarrow I_{1} > I_{3} ∴ I_{1} > I_{3} > I_{2}$

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Similar questions

Q. Consider

I_{1} = π / 4 \int 0 e^{x^{2}} d x, I_{2} = π / 4 \int 0 e^{x} d x,

I_{3} = π / 4 \int 0 e^{x^{2}} cos x d x, I_{4} = π / 4 \int 0 e^{x^{2}} sin x d x

Statement

1 : I_{2} > I_{1} > I_{3} > I_{4}

Statement

2 :

For

x \in (0, \frac{π}{4}), x > x^{2}

and

cos x > sin x

Q. If

a_{1}, a_{2}

and

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are the three values of

a

which satisfy the equation

π / 2 \int 0 (sin x + a cos x)^{3} d x - \frac{4 a}{π - 2} π / 2 \int 0 x cos x d x = 2,

then the value of

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Q. If

y = s i n (s i n x) a n d \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} t a n x + f (x) = 0, t h e n f (x) e q u a l s

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If $y = c o s^{2} x, t h e n \frac{d y}{d x} = ?$

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Geometric Interpretation of Def.Int as Limit of Sum

Standard XII Mathematics

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If I1=π/2∫0cos(sinx)dx,I2=π/2∫0sin(cosx)dx and I3=π/2∫0cosxdx, then

The correct option is A I1>I3>I2For x∈(0,π2), we know that ⇒cosx>sin(cosx) [∵f(x)=cosx−sin(cosx) is decreasing in given interval]⇒I3>I2 Also, x>sinx⇒cos(sinx)>cosx[∵cosx is decreasing function]⇒I1>I3∴I1>I3>I2

If $I_{1} = π / 2 \int 0 cos (sin x) d x, I_{2} = π / 2 \int 0 sin (cos x) d x$ and $I_{3} = π / 2 \int 0 cos x d x$ , then