Menu Close

prove-that-0-e-x-ln-1-x-sin-x-x-dx-pi-8-2-ln-2-m-n-

Tinku Tara 0 Comments
Pin




Question Number 152494 by mnjuly1970 last updated on 28/Aug/21
prove that :: Ω := ∫_(0 ) ^( ∞) (( e^( −x) .ln ((( 1)/( x)) ) sin ( x ))/(x )) dx = (( π)/( 8)) ( 2 γ +ln (2 ) ) ...■ m.n
$$ \\ $$$$\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}\:} ^{\:\infty} \frac{\:\:{e}^{\:−{x}} .\mathrm{ln}\:\left(\frac{\:\mathrm{1}}{\:{x}}\:\right)\:{sin}\:\left(\:{x}\:\right)}{{x}\:}\:{dx}\:=\:\frac{\:\pi}{\:\mathrm{8}}\:\left(\:\mathrm{2}\:\gamma\:+\mathrm{ln}\:\left(\mathrm{2}\:\right)\:\right)\:…\blacksquare\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:{m}.{n} \\ $$$$ \\ $$
Answered by Olaf_Thorendsen last updated on 31/Aug/21
Ω = ∫_0 ^∞ ((e^(−x) ln((1/x))sinx)/x) dx Ω = −Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!))∫_0 ^∞ ((x^(2n+1) e^(−x) lnx)/x) dx Ω = −Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!))∫_0 ^∞ x^(2n) e^(−x) lnx dx Ω = −Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!))ψ(2n+1)Γ(2n+1) Ω = −Σ_(n=0) ^∞ (((−1)^n )/(2n+1))ψ(2n+1) to be continued...
$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{x}} \mathrm{ln}\left(\frac{\mathrm{1}}{{x}}\right)\mathrm{sin}{x}}{{x}}\:\:{dx} \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{2}{n}+\mathrm{1}} {e}^{−{x}} \mathrm{ln}{x}}{{x}}\:\:{dx} \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}{n}} {e}^{−{x}} \mathrm{ln}{x}\:\:{dx} \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\psi\left(\mathrm{2}{n}+\mathrm{1}\right)\Gamma\left(\mathrm{2}{n}+\mathrm{1}\right) \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}{n}+\mathrm{1}}\psi\left(\mathrm{2}{n}+\mathrm{1}\right) \\ $$$${to}\:{be}\:{continued}… \\ $$$$ \\ $$

Pin

Leave a Reply

Your email address will not be published.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