Integrate[Sqrt[1/x-1],x]

%/.x->1

Limit[%%,x->0]

Integrate[Sqrt[1/x-1],{x,0,1}]

Integrate[Sqrt[1/x-1],{x,x0,1},Assumptions->0<x0<1]

Series[%,{x0,0,2}]//PowerExpand

wynik1=Integrate[Exp[-En/kT]*Exp[-Sqrt[Eg/En]],{En,0,Infinity},Assumptions->kT>0&&Eg>0]

Block[{kT=1,Eg=1},Plot[{Exp[-En/kT],Exp[-Sqrt[Eg/En]],Exp[-En/kT]*Exp[-Sqrt[Eg/En]],Exp[-((3 (Eg^(1/3)))/((2^(2/3)) (kT^(1/3))))-((3 (((En-(((Eg^(1/3)) (kT^(2/3)))/(2^(2/3)))))^2))/(2 (2^(1/3)) (Eg^(1/3)) (kT^(5/3))))]},{En,0,6},PlotRange->{0,1/2}]]

D[Exp[-En/kT]*Exp[-Sqrt[Eg/En]],En]

(Eg/(2 (Sqrt[(Eg/En)]) (En^2)))-(1/kT)//PowerExpand

Solve[((Sqrt[Eg])/(2 (En^(3/2))))-(1/kT)==0,En]//PowerExpand

Series[-En/kT-Sqrt[Eg/En],{En,(1/((2^(2/3)) ((((1/((Sqrt[Eg]) kT))))^(2/3)))),2}]//Normal//PowerExpand

-((Eg^(1/3))/((2^(2/3)) (kT^(1/3))))-(((2^(1/3)) (Eg^(1/3)))/(kT^(1/3)))//Simplify

Integrate[Exp[-((3 (Eg^(1/3)))/((2^(2/3)) (kT^(1/3))))-((3 (((En-(((Eg^(1/3)) (kT^(2/3)))/(2^(2/3)))))^2))/(2 (2^(1/3)) (Eg^(1/3)) (kT^(5/3))))],{En,-Infinity,Infinity},Assumptions->kT>0&&Eg>0]

PowerExpand[%]

Block[{Eg=EulerGamma},Plot[{(((2^(2/3)) (E^(-((3 (Eg^(1/3)))/((2^(2/3)) (kT^(1/3)))))) (Eg^(1/6)) (kT^(5/6)) (Sqrt[π]))/(Sqrt[3])),((kT MeijerG[{{},{}},{{0,(1/2),1},{}},(Eg/(4 kT))])/(Sqrt[π]))},{kT,0,3Eg},PlotRange->{0,All}]]