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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
Icansolvetheintegral∫ω1/xdxbutIcannotsolvethelimit.itseemsL=1
Commented by MJS last updated on 28/Apr/20
∫ω1/xdx=[byparts]=ω1/xx+lnω∫ω1/xxdx=∫ω1/xxdx=[t=1x→dx=−x2dt]=−∫ωttdt=Ei(tlnω)=Eilnωx=ω1/xx−lnωEilnωx+C1ω∫ω1ω1/xdx=1ω[ω1/xx−lnωEilnωx]1ω==1ω(ω1/ω+1−ω−lnωEilnωω+lnωEilnω)=ω1ω−1+lnωω(Eilnω−Eilnωω)limω→∞ω1ω−1=0limω→∞lnωω(Eilnω−Eilnωω)=?myknowledgeendshere
Answered by ~blr237~ last updated on 28/Apr/20
assumingthatw→∞,w⩾1⇒w>0letstatet=w1xthenx=1logw(t)=lnwlntf(w)=∫1ww1xdx=∫ww1wtd(lnwlnt)bypartf(w)=lnw([tlnt]ww1w−∫ww1wdtlnt)f(w)=lnw([w1wlnww−wlnw]−li(w1w)+li(w))f(w)=w1w+1−w−li(w1w)lnw+li(w)lnw1wf(w)=w1w−1−li(w1w)lnww+li(w)lnwwwehavelimw→∞w1w=1andlimw→∞li(w)lnww=1SirmrWassumingyouransweriscorrectHowcanweprovethatlimw→∞li(w1w)lnww=0knowingthatlimx→1li(x)=−∞withli(x)=∫0xdtlntcanweprovethatlimx→1li(x)ln(x)=0??
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