Evaluate-0-sin-x-1-3-log-1-x-x-dx-

Question Number 144351 by mnjuly1970 last updated on 24/Jun/21

E v a l u a t e ::

\boldsymbol ϕ := \int_{0}^{\infty} \frac{s i n \sqrt[3]{x}) l o g (\frac{1}{x})}{x} d x = ?

Answered by Dwaipayan Shikari last updated on 24/Jun/21

\int_{0}^{\infty} \frac{s i n (u)}{u^{α}} = \frac{π}{2 Γ (α) s i n (\frac{π}{2} α)}

\int_{0}^{\infty} \frac{s i n (u)}{u^{α}} l o g (\frac{1}{u}) d u = - \frac{π Γ^{'} (α)}{2 Γ {(α)}^{2} s i n (\frac{π}{2} α)} - \frac{π c o s e c (\frac{π α}{2}) c o t (\frac{π α}{2})}{2 Γ (α)}

H e r e

\int_{0}^{\infty} \frac{s i n (\sqrt[3]{x})}{x} l o g (\frac{1}{x}) d x x = u^{3}

\int_{0}^{\infty} \frac{s i n (u)}{u} l o g (\frac{1}{u}) d u = \frac{π γ}{2}

Commented by mnjuly1970 last updated on 24/Jun/21

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