How to collect q[t] from the following integration

Question

As shown in the following program, the q[t] in a can be collected from the integration by defining the integration of fe1[x] as Infe1[x], the same is true for fe2[x] in b. But for the equation c, how can I collect q[t] from it?

fe1 /: Integrate[fe1[x_], x_] := Infe1[x];
fe2 /: Integrate[fe2[x_], x_] := Infe2[x];

a = Integrate[fe1[x]*q[t], {x, 0, L1}]

b = Integrate[fe2[x]*q[t], {x, 0, L1}]   

c = Integrate[fe1[x]*fe2[x]*q[t], {x, 0, L1}]

Answer 1

You could do it with a rule:

myrule = Integrate[Times[x_, q[t]], y_] :> q[t] Integrate[x, y]

c = Integrate[fe1[x]*fe2[x]*q[t], {x, 0, L1}] /. myrule

(*  ==>  Integrate[fe1[x]*fe2[x], {x, 0, L1}]*q[t]   *)

Edit:

For more generality, you could do

myrule2 = Integrate[Times[x_, c_], {y_, y1_, y2_}] /; FreeQ[c, y] :> c  Integrate[x, {y, y1, y2}];

Integrate[(1 + Exp[h[t]]^2) g[x, t] f[x] Exp[q[t]], {x, 0, L1}] //. myrule2

(* ==> E^q[t]*(1 + E^(2*h[t])) * Integrate[f[x]*g[x, t], {x, 0, L1}] *)

Note the use of ReplaceRepeated (//.) rather than just ReplaceAll (/.).