# y := t; x := t^4; dx := diff(x,t); dy := diff(y,t); ans1 := Int(sin(x)*dx,t=-1..1)=int(sin(x)*dx,t=-1..1); clean := proc() global x,y,z,dx,dy,dz; x :='x': y:='y' : z:= 'z': dx := 'dx': dy := 'dy' : dz := 'dz': end: clean(): Int(y*z*dx + x*z*dy + x*y*dz,t); with(linalg): r := evalm([0,0,0]+t*([1,1,2]-[0,0,0])); x:=r[1]; y:=r[2]; z:=r[3]; dx:=diff(x,t);dy:=diff(y,t);dz:=diff(z,t); I1 := Int(y*z*dx + x*z*dy + x*y*dz,t=0..1) = int(y*z*dx + x*z*dy + x*y*dz,t=0..1); r := evalm([1,1,2]+t*([2,-1,2]-[1,1,2])); x:=r[1]; y:=r[2]; z:=r[3]; dx:=diff(x,t);dy:=diff(y,t);dz:=diff(z,t); I2 := Int(y*z*dx + x*z*dy + x*y*dz,t=0..1) = int(y*z*dx + x*z*dy + x*y*dz,t=0..1); r:=evalm([2,-1,2]+t*[-2,1,-2]); x:=r[1]; y:=r[2]; z:=r[3]; dx:=diff(x,t);dy:=diff(y,t);dz:=diff(z,t); I3 := Int(y*z*dx + x*z*dy + x*y*dz,t=0..1) = int(y*z*dx + x*z*dy + x*y*dz,t=0..1); ans2 := I1+I2+I3; r :=evalm([t-sin(t),1-cos(t)]); x:=r[1];y:=r[2];dx:=diff(x,t);dy:=diff(y,t); ans3 := Int(x*dx+(y+2)*dy,t=0..2*Pi) = int(x*dx+(y+2)*dy,t=0..2*Pi); lint := proc(F,r,range) local t,a,b,P,Q,R,dx,dy,dz,Li,li,f; global x,y,z; t := eval(op(1,range)); a := op(1,op(2,range)); b := op(2,op(2,range)); x := r[1]; dx := diff(x,t); y := r[2]; dy := diff(y,t); if nops(linalg[evalm](F)) = 2 then z := 0; dz := 0; R := 0 else z := r[3]; dz := diff(z,t); R := F[3] fi; Q := F[2]; P := F[1]; f := eval(P*dx+Q*dy+R*dz); Li := Int(P*'dx'+Q*'dy'+R*'dz',oes); li := int(f,t = a .. b); x := 'x'; y := 'y'; z := 'z'; RETURN(Li = li) end; lint([x,y+2,0],[t-sin(t),1-cos(t),0],t=0..2*Pi); r := [2-2*t,t-1,2-2*t]: F := [y*z,x*z,x*y]: lint(F,r,t=0..1); Lint := proc(F,r,range) local t; t := op(1,range); int(linalg[innerprod](F(op(r(t))),map(diff,r(t),t)),range) end; F := unapply(F,x,y,z); r := unapply(r,t); Lint(F,r,t=a..b); Lint(F,r,t=0..1);