Menu Close

If-0-ln-2-x-sin-x-x-dx-prove-that-4-2-pi-3-3-m-n-




Question Number 155237 by mnjuly1970 last updated on 27/Sep/21
    If   Ω = ∫_0 ^( ∞) (( ln^( 2) (x ).sin((√(x )) ))/x) dx       prove that :                   Ω = 4 γ^( 2)  + (π^( 3) /3)     ■ m.n
$$ \\ $$$$\:\:\mathrm{I}{f}\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}^{\:\mathrm{2}} \left({x}\:\right).{sin}\left(\sqrt{{x}\:}\:\right)}{{x}}\:{dx} \\ $$$$\:\:\:\:\:{prove}\:{that}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega\:=\:\mathrm{4}\:\gamma^{\:\mathrm{2}} \:+\:\frac{\pi^{\:\mathrm{3}} }{\mathrm{3}}\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$
Answered by phanphuoc last updated on 27/Sep/21
put y=(√x)→Ω=8∫_0 ^∞ ((ln^2 ysiny)/y)dy  Ω(α)=∫_0 ^∞ x^(−α) sinxdx  mellin transform M(sinx)=Γ(α)sin((πα)/2)=Ω(α)  →Ω=d^2 (Ω(α))/dα^2 =......
$${put}\:{y}=\sqrt{{x}}\rightarrow\Omega=\mathrm{8}\int_{\mathrm{0}} ^{\infty} \frac{{ln}^{\mathrm{2}} {ysiny}}{{y}}{dy} \\ $$$$\Omega\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} {x}^{−\alpha} {sinxdx} \\ $$$${mellin}\:{transform}\:{M}\left({sinx}\right)=\Gamma\left(\alpha\right){sin}\frac{\pi\alpha}{\mathrm{2}}=\Omega\left(\alpha\right) \\ $$$$\rightarrow\Omega={d}^{\mathrm{2}} \left(\Omega\left(\alpha\right)\right)/{d}\alpha^{\mathrm{2}} =…… \\ $$

Leave a Reply

Your email address will not be published.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