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Question Number 155237 by mnjuly1970 last updated on 27/Sep/21

I f Ω = \int_{0}^{\infty} \frac{{l n}^{2} (x) . s i n (\sqrt{x})}{x} d x

p r o v e t h a t :

Ω = 4 γ^{2} + \frac{π^{3}}{3} m . n

Answered by phanphuoc last updated on 27/Sep/21

p u t y = \sqrt{x} \to Ω = 8 \int_{0}^{\infty} \frac{{l n}^{2} y s i n y}{y} d y

Ω (α) = \int_{0}^{\infty} x^{- α} s i n x d x

m e l l i n t r a n s f o r m M (s i n x) = Γ (α) s i n \frac{π α}{2} = Ω (α)

\to Ω = d^{2} (Ω (α)) / d α^{2} = \dots \dots