How could I obtain real numbers for the following integration instead of imaginary ones?

Question

As shown in the following codes,

lamda1 = 1.875; L = 0.2; bL = 0.015; hL = 0.005; rouL = 2550; mL = rouL*bL*hL*L; roL =
mL/L; gama1 = (Cosh[lamda1] + Cos[lamda1])/(Sinh[lamda1] + Sin[lamda1]); 
phi1 = Cosh[lamda1*(x/L)] - Cos[lamda1*(x/L)] - gama1*(Sinh[lamda1*(x/L)] - Sin[lamda1*
(x/L)]); 
fe1 = Cosh[lamda1*(sigmma/L)] - Cos[lamda1*(sigmma/L)] - gama1*(Sinh[lamda1*(sigmma/L)] -
Sin[lamda1*(sigmma/L)]); z1 = D[fe1, sigmma]; z21 = z1^2; 
fec1 = -0.5*Integrate[z21, {sigmma, 0, x}]; ux = fec1*q4[t]^2; uy = phi1*q4[t]; uf = 
{{ux}, {uy}}; AL = {{Cos[q3[t]], -Sin[q3[t]]}, {Sin[q3[t]], Cos[q3[t]]}}; r0 = {{q1[t]},
{q2[t]}}; 
u0 = {{x}, {0}}; RL = r0 + AL . (u0 + uf); REy = Take[RL /. x -> L, -1]; vRL = D[RL, t];
Print["vRL=", MatrixForm[vRL]]; intTL = ExpandAll[Transpose[vRL] . vRL]; Print["intTL=",
intTL]; 
FullSimplify[Re[Integrate[intTL, {x, 0, L}]], Assumptions -> Element[{q1[t], q2[t], 
q3[t], q4[t], Derivative[1][q1][t], Derivative[1][q2][t], Derivative[1][q3][t],
     Derivative[1][q4][t]}, Reals]]

How could I get real numbers instead of imaginary ones?

Answer 1

You deleted your previous code. Here is a way to deal with the current one:

Chop@FullSimplify[Integrate[intTL, {x, 0, L}], Assumptions -> 
  Element[{q1[t], q2[t], q3[t], q4[t], Derivative[1][q1][t], Derivative[1]q2][t],
                                       Derivative[1][q3][t], Derivative[1][q4][t]}, Reals]]

--- The proposed solution to your previous equation --

Probably

FullSimplify[
 Re@Integrate[-7.79*q4[t]^2*Sin[18.75*x]*Sin[q3[t]]^2*Sinh[18.75*x]* Derivative[1][q4][t]^2 - 
    104*Cos[9.375*x]*Cos[q3[t]]^2*q4[t]^2*Sinh[9.375*x]*Sinh[18.75*x]* Derivative[1][q4][t]^2 - 
    104*Cos[9.375*x]*q4[t]^2*Sin[q3[t]]^2*Sinh[9.375*x]*Sinh[18.75*x]* Derivative[1][q4][t]^2 + 
    13*Cos[q3[t]]^2*q4[t]^2*Sinh[18.75*x]^2*Derivative[1][q4][t]^2 + 
    13*q4[t]^2*Sin[q3[t]]^2*Sinh[18.75*x]^2*Derivative[1][q4][t]^2, {x, 0, 1}], 
 Assumptions -> Element[{q3[t], q4[t], Derivative[1][q4][t]}, Reals]]

(* -> (1.67588*10^15 - 7.90059*10^6 Cos[2 q3[t]]) q4[t]^2 Derivative[1][q4][t]^2 *)