I1L[x_,y_]:=Integrate[Sin[t]^(2 x-1) Cos[t]^(2 y-1),{t,0,Pi/2}]

I1R[x_,y_]:=1/2 Gamma[x] Gamma[y]/Gamma[x+y]

Table[I1L[x,y],{x,1,5},{y,1,5}]

Table[I1R[x,y],{x,1,5},{y,1,5}]

Table[Integrate[Sin[t]^(k),{t,0,Pi}] // Simplify,{k,1,5}]

Table[FullSimplify[Sqrt[Pi] Gamma[(k+1)/2]/Gamma[(k+2)/2]],{k,1,5}]

(* Solid angle in dd dimensions *)

SA[dd_]:=1/(dd/2-1)! 2 Pi^(dd/2)

SA[dd] Integrate[l^(dd-1)/(l^2+a^2)^A,{l,0,Infinity}]

((((((1/(a^2))))^(-dd/2)) ((((a^2)))^(-A)) (π^(dd/2)) Gamma[A-(dd/2)] Gamma[(dd/2)])/((-1+(dd/2))! Gamma[A])) // FullSimplify

Pi^(dd/2) Gamma[A-dd/2]/Gamma[A] 1/(a^2)^(A-dd/2)

Ramondtadpole=1/2 (-lam) (mu^2)^(2-om)SA[2 om]/(2 Pi)^(2 om)Integrate[ l^(2 om-1)/(l^2+m^2),{l,0,Infinity}]

Series[(-(((2^(-1-2 om)) lam ((((1/(m^2))))^(1-om)) ((((mu^2)))^(2-om)) (π^(1-om)) Csc[om π])/((-1+om)!)) /. om-> 2 - eps),{eps,0,0}] // Simplify

FP2[a1_,a2_]:=Gamma[a1+a2]/(Gamma[a1] Gamma[a2])Integrate[DiracDelta[1-x1-x2] x1^(a1-1) x2^(a2-1)/(D1 x1 +D2 x2)^(a1+a2),{x1,0,1},{x2,0,1}]

FP2[1,1]

SA[2 om]/(2 Pi)^(2 om) Integrate[l^(2 om-1)/(l^2+m^2+p^2 x(1-x))^2,{l,0,Infinity}]

Series[(-(((2^(-2 om)) (-1+om) (π^(1-om)) ((((1/((m^2)-(p^2) (-1+x) x))))^(2-om)) Csc[om π])/((-1+om)!))/. om-> (2-eps)),{eps,0,0}]

Integrate[Log[1+4/a x(1-x)],{x,0,1}]

-2+2 (Sqrt[-1-a]) ArcTan[(1/(Sqrt[-1-a]))] // TrigToExp