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Question Number 159540 by mnjuly1970 last updated on 18/Nov/21

$$$$$$$$$$provethat:$$$$\mathit{\varphi }:={\int }_0^{\mathrm{\infty }}\frac{sin\left(\sqrt{x}\right).sin(\frac{\pi }{3}+\sqrt{x}).sin(\frac{2\pi }{3}+\sqrt{x}).ln\left(\frac{1}{x^2}\right)}{x}dx\stackrel{?}{=}\pi .(\gamma +ln\left(3\right))$$$$$$$$----------m.n$$$$$$

Answered by mindispower last updated on 19/Nov/21

$$={\int }_0^{\mathrm{\infty }}\frac{sin\left(y\right)sin(\frac{\pi }{3}+y)}{y}.2dy$$$$={\int }_0^{\mathrm{\infty }}\frac{sin\left(y\right)}{y}.\frac{1}{2}dy+{\int }_0^{\mathrm{\infty }}\frac{sin\left(y\right)cos\left(y\right)}{y}.\frac{\sqrt{3}}{2}$$$$=\frac{1}{2}.\frac{\pi }{2}+\frac{\sqrt{3}}{4}{\int }_0^{\mathrm{\infty }}\frac{sin\left(2y\right)}{2y}d\left(2y\right)$$$$=\frac{\pi }{4}+\frac{\sqrt{3}\pi }{8}$$

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