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Maths Definite Integral and Area Under Curves General Subjective Type

Published on: August 12, 2026

Prove that : 1 < dx <

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CHECK THE SOLUTION.

Sol . Let f(x) = for all x ∈

∴ f ′ (x) =

= , where g(x) = x cos x– sin x

Now, g ′ (x) = –x sin x < 0 for all x ∈ (0, π /2)

⇒ g(x) is decreasing on (0, π /2)

⇒ g(x) < g (0) for all x ∈ (0, π /2)

⇒ g(x) < 0 for all x ∈ (0, π /2)

⇒ f ′ (x) < 0 for all x ∈ (0, π /2)

⇒ f(x) is decreasing on (0, π /2)

⇒ f(x) < f(x) < f(x)

for all x ∈ (0, π /2)

⇒ < < 1

for all x ∈ (0, π /2)

⇒ dx < dx <

for all x ∈ (0, π /2)

⇒ 1 < dx <

for all x ∈ (0, π /2)

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