Problem with evaluation of integral

Question

My code is creating an error when I try to evaluate an integral. Could I ask you to reproduce this code and see what is wrong? Thank you.

Here I simply define two variables:

Find SolVm1:

odeM1 = 1/2*sigma*Vm''[x] + ((delta*a0)/a1 - a1*x)*Vm'[x] - em*Vm[x] == -((2*x - 2*lambda*ep*(a0 + a1*x))/(em - ep));
SolVm1 = DSolveValue[odeM1, Vm[x], x] // FullSimplify

Find SolVp3:

odeP3 = 1/2*sigma*Vp''[x] + ((delta*a0)/a1 - a1*x)*Vp'[x] - ep*Vp[x] == -((-2*x + 2*em*(a0 + a1*x)* lambda)/(em - ep));

SolGeneralHomoP3 = C[2] Hypergeometric1F1[ep/(2 a1), 1/2, (a1^2 x - a0 delta)^2/(a1^3 sigma)]

odeP3 /. Vp -> (h0 + h1 # &) /. Equal -> Subtract;
eqn = CoefficientList[%, x];
solh = Solve[eqn == 0, {h0, h1}][[1]]
Simplify[odeP3 /. Vp -> (h0 + h1 # &) /. solh]
SolParticularNonHomoP3 = h0 + h1 x /. solh

SolVp3 = SolGeneralHomoP3 + SolParticularNonHomoP3 // FullSimplify

From here on is where I make the mistake:

Rename the constant in SolVp3 and transform SolVm1 and SolVp3 into functions:

Vm[x_] = SolVm1 /. Integrate -> Inactive[NIntegrate];
Vp[x_] = SolVp3 /. C[2] -> C[3];

Find the constants:

Jx[x_] = em*Vm[x] + ep*Vp[x];
Jxx[x_] = em*D[Vm[x], x] + ep*D[Vp[x], x];
Jpi[x_] = Vm[x] + Vp[x];
Jpipi[x_] = D[Vm[x], x] + D[Vp[x], x];

constants = Solve[{Jx[xhat] == 0, Jxx[xhat] == 0, Jpi[xhat] == (2*theta)/((epsilon - 1)*(1 + curlyphi))*xhat}, {C[1], C[2], C[3]}];

SolJx[x_] = Jx[x] /. constants[[1]];
SolJxx[x_] = Jxx[x] /. constants[[1]];
SolJpi[x_] = Jpi[x] /. constants[[1]];
SolJpipi[x_] = Jpipi[x] /. constants[[1]];

Specify parameters:

delta = 2/100;
epsilon = 3/2;
curlyphi = 5;
xi = 5/1000;
lambda = 1/2;
theta = lambda*(epsilon - 1)*(1 + curlyphi) + 0.05;
sigma = (20/100)^2;
muA = 5/100;
sigmaA = 10/100;
rn = delta + muA - sigmaA^2;
em = (delta*theta - Sqrt[delta^2*theta^2 + 4*theta*(epsilon - 1)*(1 + curlyphi)])/(2*theta);
ep = (delta*theta + Sqrt[delta^2*theta^2 + 4*theta*(epsilon - 1)*(1 + curlyphi)])/(2*theta);
a1 = (-delta*theta + Sqrt[delta^2*theta^2 + 4*theta*(epsilon - 1)*(1 + curlyphi)])/(2*theta);
a0 = -((xi + rn - 1/2*sigma - sigmaA*Sqrt[sigma])/(delta + a1))*a1;
a = (-delta*theta + Sqrt[delta^2*theta^2 + 4*lambda*(epsilon - 1)^2*(1 + curlyphi)^2])/(2*lambda*(epsilon - 1)*(1 + curlyphi));

Find xhat (ultimate goal):

Solxhat = NSolve[Activate[SolJpipi[xhat]] == (2*theta)/((epsilon - 1)*(1 + curlyphi)), xhat]

The error:

What I don't understand: why does this error arise? I am setting x=xhat, this should not affect the variable of integration but only the limit.

I can notice something is wrong because this does not compute:

Activate[SolJx[-0.1]]

Answer 1

Maybe so:

 delta = 2/100;
 epsilon = 3/2;
 curlyphi = 5;
 xi = 5/1000;
 lambda = 1/2;
 theta = lambda*(epsilon - 1)*(1 + curlyphi) + 5/100;
 sigma = (20/100)^2;
 muA = 5/100;
 sigmaA = 10/100;
 rn = delta + muA - sigmaA^2;
 em = (delta*theta - 
 Sqrt[delta^2*theta^2 + 4*theta*(epsilon - 1)*(1 + curlyphi)])/(2*
 theta);
 ep = (delta*theta + 
 Sqrt[delta^2*theta^2 + 4*theta*(epsilon - 1)*(1 + curlyphi)])/(2*
 theta);
 a1 = (-delta*theta + 
 Sqrt[delta^2*theta^2 + 4*theta*(epsilon - 1)*(1 + curlyphi)])/(2*
 theta);
 a0 = -((xi + rn - 1/2*sigma - sigmaA*Sqrt[sigma])/(delta + a1))*a1;
 a = (-delta*theta + 
 Sqrt[delta^2*theta^2 + 
   4*lambda*(epsilon - 1)^2*(1 + curlyphi)^2])/(2*
 lambda*(epsilon - 1)*(1 + curlyphi));

 odeM1 = 1/2*sigma*Vm''[x] + ((delta*a0)/a1 - a1*x)*Vm'[x] - 
 em*Vm[x] == -((2*x - 2*lambda*ep*(a0 + a1*x))/(em - ep)) // 
 FullSimplify // Expand

 eq1 = B + F*x + G*x + H*Vm'[x] + A*x*Vm'[x] == A*Vm[x] + J*Vm''[x] + L*Vm''[x](*Simplify equation*)

 Sol = DSolveValue[eq1, Vm[x], x] // FullSimplify(*Simple General solution without Integrals*)

 (*1/(2 A^3 (J + L)) (A^(3/2)  Sqrt[
J + L]  (2  Sqrt[
   A]  (B Sqrt[J + L] + F Sqrt[J + L] x + G Sqrt[J + L] x + 
     Sqrt[2]  Sqrt[A]  (H + A x)  C[1] + 
     A  E^((H + A x)^2/(2 A (J + L)))  Sqrt[J + L]  C[2]) - 
  Sqrt[2 \[Pi]]  (E^((H + A x)^2/(
      2 A (J + L))) (F + G) (J + L) Erf[(H + A x)/(
       Sqrt[2] Sqrt[A] Sqrt[J + L])] + 
     A  (H + A x)  C[
      2]  Erfi[(H + A x)/(Sqrt[2] Sqrt[A] Sqrt[J + L])])) + (F + 
  G) (H + A x)^3 HypergeometricPFQ[{1, 1}, {3/2, 2}, (H + A x)^2/(
 2 A (J + L))])*)

Answer 2

K[1] is a localized variable of integration.

Its occurrence is a proof, that an intermediate result contains an unresolved integral. mostly eg

f=InverseFunction[NIntegrate[expr,{K[1],0,K[2]]]

The easiest way to locate it: name and Print every line by (res=(procedure); Print[res]; res)