Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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...

$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{x}} \mathrm{ln}\left(\frac{\mathrm{1}}{{x}}\right)\mathrm{sin}{x}}{{x}}\:\:{dx} \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{2}{n}+\mathrm{1}} {e}^{−{x}} \mathrm{ln}{x}}{{x}}\:\:{dx} \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}{n}} {e}^{−{x}} \mathrm{ln}{x}\:\:{dx} \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\psi\left(\mathrm{2}{n}+\mathrm{1}\right)\Gamma\left(\mathrm{2}{n}+\mathrm{1}\right) \\ $$$$\Omega\:=\:−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}{n}+\mathrm{1}}\psi\left(\mathrm{2}{n}+\mathrm{1}\right) \\ $$$${to}\:{be}\:{continued}... \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com