Question

If I₁ = , I₂ = $\int _ { 0 } ^ { \pi / 2 } \sin ( \cos x ) d x$, I₃ = $\int _ { 0 } ^ { \pi / 2 } \cos x d x$ , then

• A. I₁> I₂> I₃
• B. I₃> I₂> I₁
• C. I₁> I₃>I₂
• D. I₃> I₁> I₂

Answer 1

Answer: (C)

I₁> I₃>I₂

Answer 2

Q sin x < x " x > 0

̃ sin (cos x) < cos x for 0 < x < p/2

̃$\int _ { 0 } ^ { \pi / 2 } \sin ( \cos x ) d x$ < ̃ I₂ < I₃ .... (i)

Further x Î [0, p/2] ̃ sin x < x and x₁, x₂ Î [0, p/2]

x₁ > x₂ ̃ cos x₁ < co s x₂

\ cos x < cos (sin x)

= $\int _ { 0 } ^ { \pi / 2 } \cos x d x$ < $\int _ { 0 } ^ { \pi / 2 } \cos x ( \sin x ) d x$

̃ I₃ < I₁ ..... (ii) (i), (ii)

̃ I₂ < I₃ < I₁ ̃ I₁ > I₃ > I₂