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Question Number 82446 by M±th+et£s last updated on 21/Feb/20
show that  ∫_0 ^∞ x^(−log(x))  x log(x) dx=e(√π)
$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{−{log}\left({x}\right)} \:{x}\:{log}\left({x}\right)\:{dx}={e}\sqrt{\pi} \\ $$
Commented by abdomathmax last updated on 21/Feb/20
changement logx=t give x=e^t  ⇒  I= ∫_(−∞) ^(+∞)  (e^t )^(−t) e^t t e^t  dt  =∫_(−∞) ^(+∞) t e^(−t^2 +2t)  dt  =∫_(−∞) ^(+∞)  t e^(−(t^2 −2t +1−1)) dt  =∫_(−∞) ^(+∞)  t e^(−(t−1)^2 +1)  dt =_(t−1=u)   e∫_(−∞) ^(+∞) (u+1)e^(−u^2 ) du  but  ∫_(−∞) ^(+∞) (u+1)e^(−u^2 ) du  =∫_(−∞) ^(+∞)  u e^(−u^2 ) du +∫_(−∞) ^(+∞)  e^(−u^2 ) du  =[−(1/2)e^(−u^2 ) ]_(−∞) ^(+∞)  +(√π)=0+(√π) ⇒  I =e(√π)
$${changement}\:{logx}={t}\:{give}\:{x}={e}^{{t}} \:\Rightarrow \\ $$$${I}=\:\int_{−\infty} ^{+\infty} \:\left({e}^{{t}} \right)^{−{t}} {e}^{{t}} {t}\:{e}^{{t}} \:{dt} \\ $$$$=\int_{−\infty} ^{+\infty} {t}\:{e}^{−{t}^{\mathrm{2}} +\mathrm{2}{t}} \:{dt}\:\:=\int_{−\infty} ^{+\infty} \:{t}\:{e}^{−\left({t}^{\mathrm{2}} −\mathrm{2}{t}\:+\mathrm{1}−\mathrm{1}\right)} {dt} \\ $$$$=\int_{−\infty} ^{+\infty} \:{t}\:{e}^{−\left({t}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}} \:{dt}\:=_{{t}−\mathrm{1}={u}} \:\:{e}\int_{−\infty} ^{+\infty} \left({u}+\mathrm{1}\right){e}^{−{u}^{\mathrm{2}} } {du} \\ $$$${but}\:\:\int_{−\infty} ^{+\infty} \left({u}+\mathrm{1}\right){e}^{−{u}^{\mathrm{2}} } {du} \\ $$$$=\int_{−\infty} ^{+\infty} \:{u}\:{e}^{−{u}^{\mathrm{2}} } {du}\:+\int_{−\infty} ^{+\infty} \:{e}^{−{u}^{\mathrm{2}} } {du} \\ $$$$=\left[−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−{u}^{\mathrm{2}} } \right]_{−\infty} ^{+\infty} \:+\sqrt{\pi}=\mathrm{0}+\sqrt{\pi}\:\Rightarrow \\ $$$${I}\:={e}\sqrt{\pi} \\ $$
Commented by M±th+et£s last updated on 21/Feb/20
thank you sir
$${thank}\:{you}\:{sir} \\ $$
Commented by mathmax by abdo last updated on 21/Feb/20
you are welcome.
$${you}\:{are}\:{welcome}. \\ $$

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