Let ℕ be the set of natural numbers. For n ∈ ℕ , define I n = ∫ 0 π x sin 2 n ( x ) sin 2 n ( x ) + cos 2 n ( x ) d x . Then for m , n ∈ ℕ

Question

Let ℕ be the set of natural numbers. For n ∈ ℕ , define I n = ∫ 0 π x sin 2 n ( x ) sin 2 n ( x ) + cos 2 n ( x ) d x . Then for m , n ∈ ℕ, I m I n for all m n, I m > I n for all m n, I m = I n for all m ≠ n, I m I n for some m n and I m > I n for some m n

MARKS App · Accepted Answer

I n = 1 2 ∫ 0 π x sin 2 n x sin 2 n x + cos 2 n x + ( π - x ) sin 2 n x sin 2 n x + cos 2 n x d x = π 2 ∫ 0 π sin 2 n x d x sin 2 n x + cos 2 n x = 2 × π 2 ∫ 0 π / 2 sin 2 n x d x sin 2 n x + cos 2 n x = π 2 ∫ 0 π / 2 sin 2 n x + cos 2 n x sin 2 n x + cos 2 n x d x = π 2 × π 2 = π 2 4 ⇒ I m = I n ∀ m , n

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