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Question Number 134102 by mnjuly1970 last updated on 27/Feb/21
          prove  that :   𝛗=∫_0 ^( ∞) ((cos((√x) ))/(e^(2Ο€(√x) ) βˆ’1))dx=1βˆ’(e/((eβˆ’1)_ ^2 ))
provethat:Ο•=∫0∞cos(x)e2Ο€xβˆ’1dx=1βˆ’e(eβˆ’1)2
Answered by Dwaipayan Shikari last updated on 27/Feb/21
x=t^2   2∫_0 ^∞ t((cos(t))/(e^(2Ο€t) βˆ’1))dt=Ξ£_(n=0) ^∞ ∫_0 ^∞ te^(βˆ’2Ο€nt+it) +te^(βˆ’2Ο€ntβˆ’it) dt  =Ξ£_(n=0) ^∞ (1/((2Ο€nβˆ’i)^2 ))+(1/((2Ο€n+i)^2 ))=(1/(4Ο€^2 ))(ψ^1 (βˆ’(i/(2Ο€)))+ψ^1 ((i/(2Ο€))))  =(1/(4Ο€^2 ))(Ο€^2 csc^2 Ο€((i/(2Ο€))))+1=1+(1/4)(((2i)/(e^((βˆ’1)/2) βˆ’e^(1/2) )))^2 =1βˆ’(e/((eβˆ’1)^2 ))
x=t22∫0∞tcos(t)e2Ο€tβˆ’1dt=βˆ‘βˆžn=0∫0∞teβˆ’2Ο€nt+it+teβˆ’2Ο€ntβˆ’itdt=βˆ‘βˆžn=01(2Ο€nβˆ’i)2+1(2Ο€n+i)2=14Ο€2(ψ1(βˆ’i2Ο€)+ψ1(i2Ο€))=14Ο€2(Ο€2csc2Ο€(i2Ο€))+1=1+14(2ieβˆ’12βˆ’e12)2=1βˆ’e(eβˆ’1)2
Commented by mnjuly1970 last updated on 27/Feb/21
bravo bravo  mr payan ...excellent...
bravobravomrpayan…excellent…

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