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find-the-value-of-0-x-ch-x-dx-




Question Number 66786 by mathmax by abdo last updated on 19/Aug/19
find the value of ∫_0 ^∞  (x/(ch(x)))dx
findthevalueof0xch(x)dx
Commented by mathmax by abdo last updated on 20/Aug/19
let I =∫_0 ^∞  (x/(ch(x)))dx ⇒ I =∫_0 ^∞ ((2x)/(e^x  +e^(−x) ))dx =∫_0 ^∞  ((2xe^(−x) )/(1+e^(−2x) ))dx  =2 ∫_0 ^∞   x e^(−x) (Σ_(n=0) ^∞ (−1)^n e^(−2nx) )dx  =2Σ_(n=0) ^∞ (−1)^n  ∫_0 ^∞ x e^(−(2n+1)x) dx =_((2n+1)x=t)    2Σ_(n=0) ^∞ (−1)^n ∫_0 ^∞ (t/(2n+1))e^(−t) (dt/(2n+1))  =2 Σ_(n=0) ^∞  (((−1)^n )/((2n+1)^2 ))∫_0 ^∞  t e^(−t)  dt    but ∫_0 ^∞  t e^(−t)  dt =Γ(2)=1!=1 ⇒  I =2 Σ_(n=0) ^∞  (((−1)^n )/((2n+1)^2 ))   ....be continued...
letI=0xch(x)dxI=02xex+exdx=02xex1+e2xdx=20xex(n=0(1)ne2nx)dx=2n=0(1)n0xe(2n+1)xdx=(2n+1)x=t2n=0(1)n0t2n+1etdt2n+1=2n=0(1)n(2n+1)20tetdtbut0tetdt=Γ(2)=1!=1I=2n=0(1)n(2n+1)2.becontinued

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