Clear[f]
f[x_, y_, nu_] = x^(nu-1)/(E^(x - y) + 1) ;

Manipulate[
Plot[f[x, y, nu],
 {x, 0, y},
PlotRange -> Full
],
{{nu, 1}, -1/2, 9/2, 1/2},
{{y, 100}, 10, 100000}
]

Module[{a, b},
a[x_, nu_] = Map[ 
f[x, #, nu]
&, { 100, 400, 600, 800}] ;
b[x_] = Flatten[Map[ a[x, #]&, {1, 3/2(*, 2, 5/2*)}]] ;

Plot[
Evaluate[b[x]]
,
 {x, 0, 1000},
PlotRange -> Full

]]

(*Clear[f, y, a, nu]
f[x_, y_, nu_] = x^(nu-1)/(E^(y -x) + 1) 
Integrate[ f[x, y, nu], {x, 0, y}]
*)
(*Manipulate[
Plot[
Integrate[
f[a, y, nu],
 {a, 0, 100},
PlotRange -> Full
],
{y, 1, 1000},
{{nu, 1}, -1/2, 9/2, 1/2}
]*)

Clear[j, i]
$Assumptions = j ∈ Integers && j > 0
i = Integrate[a^j/(E^a + 1), {a, 0, Infinity}]
i // TraditionalForm

Product[ (nu-k), {k, 1, j}]

(((-j+nu) Pochhammer[1-j+nu,j])/nu)

(*(((-j+nu) Gamma[1-j+nu +j])/(nu Gamma[1 -j + nu]))   -> (( Gamma[nu])/( Gamma[nu- j]))*)

Clear[c]
c[j_] = 2 (1 - 2^(-j)) Zeta[j+1]
Map[ c[#] &, {1, 3, 5, 7}]

c[j] // TraditionalForm

Clear[i]
$Assumptions = alpha > 0 ;
 
i = (1/Gamma[nu])Integrate[x^(nu-1)/(alpha + x), {x, 0, Infinity}] ;

i /. nu -> ν /. alpha -> α// FullSimplify //  TraditionalForm

Map[ Zeta[#] &, {3/2, 2, 5/2, 3}] // N