Question Number 103220 by Dwaipayan Shikari last updated on 13/Jul/20

∫_0 ^∞ (x^3 /(e^x +1))dx

Answered by mathmax by abdo last updated on 13/Jul/20

I =∫_0 ^∞  (x^3 /(e^x +1))dx ⇒ I =∫_0 ^∞   ((x^3  e^(−x) )/(1+e^(−x) ))dx =∫_0 ^∞  x^(3 ) e^(−x)  (Σ_(n=0) ^∞ (−1)^n  e^(−nx) )dx  =Σ_(n=0) ^∞  (−1)^(n )  ∫_0 ^∞   x^3  e^(−(n+1)x ) dx  =_((n+1)x =t) Σ_(n=0) ^∞  (−1)^n  ∫_0 ^∞ (t^3 /((n+1)^3 )) e^(−t )  (dt/((n+1)))  =Σ_(n=0) ^∞  (((−1)^n )/((n+1)^4 )) ∫_0 ^∞  t^3  e^(−t)  dt  we know  ∫_0 ^∞  t^(x−1)  e^(−t)  dt =Γ(x) ⇒  I =Γ(4)×Σ_(n=1) ^∞  (((−1)^(n−1) )/n^4 )  =−3!×Σ_(n=1) ^∞  (((−1)^n )/n^4 )  we have δ(x) =Σ_(n=1) ^∞  (((−1)^n )/n^x ) =(2^(1−x) −1)ξ(x) ⇒  Σ_(n=1) ^∞  (((−1)^n )/n^4 ) =δ(4) =(2^(−3) −1)ξ(4) =((1/8)−1)ξ(4) =−(7/8)ξ(4) ⇒  I =−(3!)×(−(7/8))ξ(4) =((3.2.7)/(4.2))ξ(4) =((21)/4)ξ(4)  if ξ(4) =(π^4 /(90)) we get  I =((21π^4 )/(360)) =((3.7π^4 )/(3.120)) ⇒I =((7π^4 )/(120))

Commented byDwaipayan Shikari last updated on 13/Jul/20

Thanking you

Commented byabdomsup last updated on 13/Jul/20

you are welcome