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Question Number 75795 by ~blr237~ last updated on 17/Dec/19

Find out  ∫_0 ^∞  ((argsh(x))/x)dx

$$\mathrm{Find}\:\mathrm{out}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{argsh}\left(\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx} \\ $$

Commented by mathmax by abdo last updated on 18/Dec/19

let A =∫_0 ^∞  ((argsh(x))/x)dx ⇒ A =∫_0 ^∞ ((ln(x+(√(1+x^2 ))))/x)dx  changement argsh(x)=t give x =sh(t)=((e^t −e^(−t) )/2) ⇒  A =∫_0 ^∞   (t/(sh(t)))ch(t)dt  =∫_0 ^∞   ((tch(t))/(sh(t)))dt  =∫_0 ^∞   t ((e^t +e^(−t) )/(e^t −e^(−t) )) dt =∫_0 ^∞  t ((1+e^(−2t) )/(1−e^(−2t) ))dt  =∫_0 ^∞  (t+te^(−2t) )(Σ_(n=0) ^∞  e^(−2nt) )dt  =Σ_(n=0) ^∞ ∫_0 ^∞  t e^(−2nt)  dt +Σ_(n=0) ^∞ ∫_0 ^∞  t e^(−(2+2n)t)  dt   by parts  ∫_0 ^∞  t e^(−2nt) dt=[−(t/(2n)) e^(−2nt) ]_0 ^(+∞)  −∫_0 ^∞  (−(1/(2n)))e^(−2nt)  dt  =(1/(2n)) ∫_0 ^∞   e^(−2nt)  dt =−(1/(4n^2 )) [e^(−2n) ]_0 ^(+∞) =(1/(4n^2 ))  ∫_0 ^∞   t e^(−(2+2n)t)  dt =_(t=(u/2))  ∫_0 ^∞   (u/2) e^(−(n+1)t)  (du/2)  =(1/4)∫_0 ^∞   u e^(−(n+1)u)  du  =_((n+1)u=z)   (1/4)∫_0 ^∞  (z/(n+1))e^(−z)  (dz/(n+1))  =(1/(4(n+1)^2 ))∫_0 ^∞  z e^(−z)  dz =(1/(4(n+1)^2 )){ [−z e^(−z) ]_0 ^∞ −∫_0 ^∞  (−e^(−z) )}  =(1/(4(n+1)^2 )){ [−e^(−z) ]_0 ^(+∞) } =(1/(4(n+1)^2 )) ⇒  A =Σ_(n=0) ^∞   (1/(4n^2 )) +Σ_(n=0) ^∞  (1/(4(n+1)^2 ))   so A is divergent ...!  or something went wrong..

$${let}\:{A}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{argsh}\left({x}\right)}{{x}}{dx}\:\Rightarrow\:{A}\:=\int_{\mathrm{0}} ^{\infty} \frac{{ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}{{x}}{dx} \\ $$$${changement}\:{argsh}\left({x}\right)={t}\:{give}\:{x}\:={sh}\left({t}\right)=\frac{{e}^{{t}} −{e}^{−{t}} }{\mathrm{2}}\:\Rightarrow \\ $$$${A}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}}{{sh}\left({t}\right)}{ch}\left({t}\right){dt}\:\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{tch}\left({t}\right)}{{sh}\left({t}\right)}{dt} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \:\:{t}\:\frac{{e}^{{t}} +{e}^{−{t}} }{{e}^{{t}} −{e}^{−{t}} }\:{dt}\:=\int_{\mathrm{0}} ^{\infty} \:{t}\:\frac{\mathrm{1}+{e}^{−\mathrm{2}{t}} }{\mathrm{1}−{e}^{−\mathrm{2}{t}} }{dt} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \:\left({t}+{te}^{−\mathrm{2}{t}} \right)\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:{e}^{−\mathrm{2}{nt}} \right){dt} \\ $$$$=\sum_{{n}=\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \:{t}\:{e}^{−\mathrm{2}{nt}} \:{dt}\:+\sum_{{n}=\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \:{t}\:{e}^{−\left(\mathrm{2}+\mathrm{2}{n}\right){t}} \:{dt}\: \\ $$$${by}\:{parts}\:\:\int_{\mathrm{0}} ^{\infty} \:{t}\:{e}^{−\mathrm{2}{nt}} {dt}=\left[−\frac{{t}}{\mathrm{2}{n}}\:{e}^{−\mathrm{2}{nt}} \right]_{\mathrm{0}} ^{+\infty} \:−\int_{\mathrm{0}} ^{\infty} \:\left(−\frac{\mathrm{1}}{\mathrm{2}{n}}\right){e}^{−\mathrm{2}{nt}} \:{dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{n}}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{nt}} \:{dt}\:=−\frac{\mathrm{1}}{\mathrm{4}{n}^{\mathrm{2}} }\:\left[{e}^{−\mathrm{2}{n}} \right]_{\mathrm{0}} ^{+\infty} =\frac{\mathrm{1}}{\mathrm{4}{n}^{\mathrm{2}} } \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{t}\:{e}^{−\left(\mathrm{2}+\mathrm{2}{n}\right){t}} \:{dt}\:=_{{t}=\frac{{u}}{\mathrm{2}}} \:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{u}}{\mathrm{2}}\:{e}^{−\left({n}+\mathrm{1}\right){t}} \:\frac{{du}}{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} \:\:{u}\:{e}^{−\left({n}+\mathrm{1}\right){u}} \:{du}\:\:=_{\left({n}+\mathrm{1}\right){u}={z}} \:\:\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} \:\frac{{z}}{{n}+\mathrm{1}}{e}^{−{z}} \:\frac{{dz}}{{n}+\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\int_{\mathrm{0}} ^{\infty} \:{z}\:{e}^{−{z}} \:{dz}\:=\frac{\mathrm{1}}{\mathrm{4}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\left\{\:\left[−{z}\:{e}^{−{z}} \right]_{\mathrm{0}} ^{\infty} −\int_{\mathrm{0}} ^{\infty} \:\left(−{e}^{−{z}} \right)\right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\left\{\:\left[−{e}^{−{z}} \right]_{\mathrm{0}} ^{+\infty} \right\}\:=\frac{\mathrm{1}}{\mathrm{4}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:\Rightarrow \\ $$$${A}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\mathrm{4}{n}^{\mathrm{2}} }\:+\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\mathrm{4}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{so}\:{A}\:{is}\:{divergent}\:...! \\ $$$${or}\:{something}\:{went}\:{wrong}.. \\ $$

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