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 174685 by mnjuly1970 last updated on 08/Aug/22

prove that : Ω = ∫_0 ^( ∞) (( x^( 2) )/(cosh(x ))) dx = (π^( 3) /( 8))

$$ \\ $$$$\:\:\:\:\:{prove}\:{that}\:: \\ $$$$\: \\ $$$$\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\:{x}^{\:\mathrm{2}} }{{cosh}\left({x}\:\right)}\:{dx}\:=\:\frac{\pi^{\:\mathrm{3}} }{\:\mathrm{8}} \\ $$$$ \\ $$

Answered by princeDera last updated on 08/Aug/22

(1/(cosh(x))) = ((2e^(−x) )/(1 + e^(−2x) )) Ω = ∫_0 ^∞ ((2x^2 e^(−x) )/(1+cosh (x)))dx = 2∫_0 ^∞ x^2 e^(−x) Σ_(k≥0) (−1)^k e^(−2kx) dx = 2Σ_(k≥0) (−1)^k ∫_0 ^∞ x^2 e^(−(2k+1)x) dx = 4Σ_(k≥0) (((−1)^k )/((2k+1)^3 )) 4B(3) = 4((π^3 /(32))) = (π^3 /8) where B(z) is the dirichlet beta function

$$ \\ $$$$ \\ $$$$ \\ $$$$\frac{\mathrm{1}}{\mathrm{co}{sh}\left({x}\right)}\:=\:\frac{\mathrm{2}{e}^{−{x}} }{\mathrm{1}\:+\:{e}^{−\mathrm{2}{x}} } \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{2}{x}^{\mathrm{2}} {e}^{−{x}} }{\mathrm{1}+\mathrm{cosh}\:\left({x}\right)}{dx}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}} \underset{{k}\geqslant\mathrm{0}} {\sum}\left(−\mathrm{1}\right)^{{k}} {e}^{−\mathrm{2}{kx}} {dx} \\ $$$$=\:\mathrm{2}\underset{{k}\geqslant\mathrm{0}} {\sum}\left(−\mathrm{1}\right)^{{k}} \int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{−\left(\mathrm{2}{k}+\mathrm{1}\right){x}} {dx}\:=\:\mathrm{4}\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{k}} }{\left(\mathrm{2}{k}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$ \\ $$$$\mathrm{4}\mathfrak{B}\left(\mathrm{3}\right)\:=\:\mathrm{4}\left(\frac{\pi^{\mathrm{3}} }{\mathrm{32}}\right)\:=\:\frac{\pi^{\mathrm{3}} }{\mathrm{8}} \\ $$$${where}\:\mathfrak{B}\left({z}\right)\:{is}\:{the}\:{dirichlet}\:{beta}\:{function} \\ $$$$ \\ $$$$ \\ $$

Commented by MME last updated on 08/Aug/22

Boss mi

$${Boss}\:{mi} \\ $$

Commented by mnjuly1970 last updated on 08/Aug/22

thanks alot sir just , β (3) =(π^( 3) /(32)) typo...

$${thanks}\:{alot}\:{sir} \\ $$$$\:{just}\:,\:\:\beta\:\left(\mathrm{3}\right)\:=\frac{\pi^{\:\mathrm{3}} }{\mathrm{32}} \\ $$$$\:\:{typo}... \\ $$

Commented by princeDera last updated on 08/Aug/22

corrected. thank you

$${corrected}. \\ $$$${thank}\:{you} \\ $$$$ \\ $$

Commented by princeDera last updated on 08/Aug/22

my boss

$$ \\ $$$${my}\:{boss} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com