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Question Number 91054 by Mikael_786 last updated on 27/Apr/20

Commented by MJS last updated on 27/Apr/20

I can solve the integral ∫ω^(1/x) dx but I cannot  solve the limit. it seems  L=1

Icansolvetheintegralω1/xdxbutIcannotsolvethelimit.itseemsL=1

Commented by MJS last updated on 28/Apr/20

∫ω^(1/x) dx=       [by parts]  =ω^(1/x) x+ln ω ∫(ω^(1/x) /x)dx=         ∫(ω^(1/x) /x)dx=            [t=(1/x) → dx=−x^2 dt]       =−∫(ω^t /t)dt=Ei (tln ω) =Ei ((ln ω)/x)    =ω^(1/x) x−ln ω Ei ((ln ω)/x) +C    (1/ω)∫_1 ^ω ω^(1/x) dx=(1/ω)[ω^(1/x) x−ln ω Ei ((ln ω)/x)]_1 ^ω =  =(1/ω)(ω^(1/ω+1) −ω−ln ω Ei ((ln ω)/ω) +ln ω Ei ln ω)  =ω^(1/ω) −1+((ln ω)/ω)(Ei ln ω −Ei ((ln ω)/ω))  lim_(ω→∞)  ω^(1/ω) −1 =0  lim_(ω→∞)  ((ln ω)/ω)(Ei ln ω −Ei ((ln ω)/ω)) =?  my knowledge ends here

ω1/xdx=[byparts]=ω1/xx+lnωω1/xxdx=ω1/xxdx=[t=1xdx=x2dt]=ωttdt=Ei(tlnω)=Eilnωx=ω1/xxlnωEilnωx+C1ωω1ω1/xdx=1ω[ω1/xxlnωEilnωx]1ω==1ω(ω1/ω+1ωlnωEilnωω+lnωEilnω)=ω1ω1+lnωω(EilnωEilnωω)limωω1ω1=0limωlnωω(EilnωEilnωω)=?myknowledgeendshere

Answered by ~blr237~ last updated on 28/Apr/20

assuming that  w→∞ ,w≥1 ⇒ w>0  let state   t=w^(1/x)   then  x=(1/(log_w (t)))=((lnw)/(lnt))   f(w)=∫_1 ^w w^(1/x) dx=∫_w ^w^(1/w)  td(((lnw)/(lnt)))   by part f(w)=lnw([(t/(lnt))]_w ^w^(1/w)  −∫_w ^w^(1/w)  (dt/(lnt)))  f(w)=lnw([(w^(1/w) /((lnw)/w)) −(w/(lnw))]−li(w^(1/w) )+li(w))  f(w)=w^((1/w)+1) −w−li(w^(1/w) )lnw+li(w)lnw  (1/w)f(w)= w^(1/w) −1−li(w^(1/w) )((lnw)/w) + li(w)((lnw)/w)   we have  lim_(w→∞)  w^(1/w) =1   and  lim_(w→∞)  li(w)((lnw)/w)=1    Sir mrW   assuming your answer is correct   How can we prove that   lim_(w→∞)  li(w^(1/w) )((lnw)/w) =0  knowing  that lim_(x→1)  li(x)=−∞  with li(x)=∫_0 ^x (dt/(lnt))      can we prove that lim_(x→1)  li(x)ln(x)=0 ??

assumingthatw,w1w>0letstatet=w1xthenx=1logw(t)=lnwlntf(w)=1ww1xdx=ww1wtd(lnwlnt)bypartf(w)=lnw([tlnt]ww1www1wdtlnt)f(w)=lnw([w1wlnwwwlnw]li(w1w)+li(w))f(w)=w1w+1wli(w1w)lnw+li(w)lnw1wf(w)=w1w1li(w1w)lnww+li(w)lnwwwehavelimww1w=1andlimwli(w)lnww=1SirmrWassumingyouransweriscorrectHowcanweprovethatlimwli(w1w)lnww=0knowingthatlimx1li(x)=withli(x)=0xdtlntcanweprovethatlimx1li(x)ln(x)=0??

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