0-sin-x-log-2-x-x-dx-pi-24-12-2-pi-2-

Question Number 138967 by mnjuly1970 last updated on 20/Apr/21

Θ = \int_{0}^{\infty} \frac{s i n (x) . {l o g}^{2} (x)}{x} d x = \frac{π}{24} (12 γ^{2} + π^{2}) \dots . ✓

Answered by Dwaipayan Shikari last updated on 20/Apr/21

χ (μ) = \int_{0}^{\infty} \frac{s i n x}{x^{μ}} d x = \frac{π}{2 Γ (μ) s i n (\frac{π μ}{2})}

χ^{″} (1) = \int_{0}^{\infty} \frac{s i n x}{x} {l o g}^{2} (x) d x = \frac{π}{2} (\frac{π^{2}}{12} + γ^{2})