1.Using: ​​
Log[Gamma[1-x]]==EulerGamma*x+Sum
Zeta[n]
n
*x^n,{n,2,Infinity}
​and ​​
Log[Gamma[1+x]]==-EulerGamma*x+Sum
Zeta[n]
n
*(-x)^n,{n,2,Infinity}
In[]:=
Integrate
EulerGamma*x+
Zeta[n]
n
*x^n
x
,{x,0,1}
Out[]=
EulerGamma+
Zeta[n]
2
n
if Re[n]>0
EulerGamma+NSum
Zeta[n]
2
n
,{n,2,Infinity}(*NOClosedform*)
Out[]=
1.41321
In[]:=
NIntegrate[Log[Gamma[1-x]]/x,{x,0,1}]
Out[]=
1.41321
(*----------------------------------------------------*)
In[]:=
Integrate
-EulerGamma*x+
Zeta[n]
n
*(-x)^n
x
,{x,0,1}
Out[]=
-EulerGamma+
n
(-1)
Zeta[n]
2
n
if Re[n]>0
In[]:=
-EulerGamma+NSum
n
(-1)
Zeta[n]
2
n
,{n,2,Infinity}(*NOClosedform*)
Out[]=
-0.256875
In[]:=
NIntegrate[Log[Gamma[1+x]]/x,{x,0,1}]
Out[]=
-0.256875
(*----------------------------------------------------*)
In[]:=
NIntegrate[(2xArcCot[x]-Log[1+1/x^2])Log[1-Exp[-2Pix]]/(2Pi(1+x^2)),{x,0,Infinity}]
Out[]=
0.235681
EulerGamma-2+
Pi^2
12
+NSum
Zeta[n]
n^2
,{n,2,Infinity}(*WithAI.NOClosedform*)
Out[]=
0.235681
(*----------------------------------------------------*)