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Question Number 40885 by prof Abdo imad last updated on 28/Jul/18
prove that  1) ∫_0 ^1  ((t^p ln(t))/(t−1))dt =(π^2 /6) −Σ_(k=1) ^p  (1/k^2 )  2) ∫_0 ^1   ((t^(2p) ln(t))/(t^2 −1))dt =(π^2 /8) −Σ_(k=0) ^(p−1)   (1/((2k+1)^2 ))
provethat1)01tpln(t)t1dt=π26k=1p1k22)01t2pln(t)t21dt=π28k=0p11(2k+1)2
Commented by math khazana by abdo last updated on 02/Aug/18
1) ∫_0 ^1   ((t^p ln(t))/(t−1)) dt =−∫_0 ^1  t^p ln(t)(Σ_(n=0) ^∞ t^n ))dt  =−Σ_(n=0) ^∞   ∫_0 ^1  t^(n+p) ln(t) dt =−Σ_(n=0) ^∞  A_n   and by  parts   A_n =∫_0 ^1  t^(n+p) ln(t)dt =[(1/(n+p+1))t^(n+p+1) ln(t)]_0 ^1   −∫_0 ^1   (1/(n+p+1)) t^(n+p)  dt  =−(1/((n+p+1)^2 )) ⇒−Σ_(n=0) ^∞  A_n =Σ_(n=0) ^∞    (1/((n+p+1)^2 ))  =_(n+p+1=k)     Σ_(k=p+1) ^∞   (1/k^2 )  but  Σ_(k=1) ^∞   (1/k^2 ) =(π^2 /6) =Σ_(k=1) ^p   (1/k^2 ) +Σ_(k=p+1) ^∞   (1/k^2 ) ⇒  Σ_(k=p+1) ^∞  =(π^2 /6) −Σ_(k=1) ^p  (1/k^2 )  = ∫_0 ^1  ((t^p ln(t))/(t−1)) dt.
1)01tpln(t)t1dt=01tpln(t)(n=0tn))dt=n=001tn+pln(t)dt=n=0AnandbypartsAn=01tn+pln(t)dt=[1n+p+1tn+p+1ln(t)]01011n+p+1tn+pdt=1(n+p+1)2n=0An=n=01(n+p+1)2=n+p+1=kk=p+11k2butk=11k2=π26=k=1p1k2+k=p+11k2k=p+1=π26k=1p1k2=01tpln(t)t1dt.

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