Clear["Global`*"];
jint[r_,k_,θ_]:=(1/2)(((E^(-2I r k ((Sin[θ/2])^2)))/r) {0,0,k}+ ((E^(2I r k ((Sin[θ/2])^2)))/r) {0,0,k}+(E^(-2I r k ((Sin[θ/2])^2)))((( k)/r)+(1/(I (r^2)))) {Sin[θ],0,Cos[θ]}+(E^(2I r k ((Sin[θ/2])^2)))((( k)/r)-(1/(I (r^2)))){Sin[θ],0,Cos[θ]});

jintav[r_,k_,θ0_,Δθ_]:=NIntegrate[(1/2)(((E^(-2I r k ((Sin[θp/2])^2)))/r)  {0,0,k}+ ((E^(2I r k ((Sin[θp/2])^2)))/r)  {0,0,k}+(E^(-2I r k ((Sin[θp/2])^2)))((( k)/r)+(1/(I (r^2)))){Sin[θp],0,Cos[θp]}+(E^(2I r k ((Sin[θp/2])^2)))((( k)/r)-(1/(I (r^2)))){Sin[θp],0,Cos[θp]})Sin[θp],{θp,θ0,θ0+Δθ}, Method->"ClenshawCurtisRule"]/Δθ;
r0=30;
k0=2;
Δθ=6(π/180);
Plot[{jint[r0,k0,θ][[1]],jintav[r0,k0,θ,Δθ][[1]]},{θ,0,π}]
Plot[{jint[r0,k0,θ][[3]],jintav[r0,k0,θ,Δθ][[3]]},{θ,0,π}]