Menu Close

advanced-math-two-simple-and-nice-integrals-prove-that-1-0-sin-e-x-ln-x-x-dx-0-2-0-sin-x-2-2-l




Question Number 122713 by mnjuly1970 last updated on 19/Nov/20
            ...  advanced  math ...       two  simple and nice integrals:     prove  that::                    Ω_1 =∫_0 ^( ∞) ((sin(e^(−γ) x)ln(x))/x) dx=0              Ω_2  =∫_0 ^( ∞) ((sin(x^((√2)/2) )ln(x))/x)dx=−πγ                note :: γ :  Euler−mascheroni                                     constant.                            .m.n.
$$\:\:\:\:\:\:\:\:\:\:\:\:…\:\:{advanced}\:\:{math}\:… \\ $$$$\:\:\:\:\:{two}\:\:{simple}\:{and}\:{nice}\:{integrals}: \\ $$$$\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({e}^{−\gamma} {x}\right){ln}\left({x}\right)}{{x}}\:{dx}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Omega_{\mathrm{2}} \:=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}^{\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}} \right){ln}\left({x}\right)}{{x}}{dx}=−\pi\gamma\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{note}\:::\:\gamma\::\:\:\mathscr{E}{uler}−{mascheroni} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{constant}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *