(*Specialcases:*)
Integrate
1
A^2*(Exp[-2Bx]-2Exp[-B*x])+2t*x+
Exp[-Bx]
B
+c
/.t->0,x(*t=0*)
Out[]=
-
-Bx

2
A
-2
2
A
Bx

+c
2Bx

Log
2
A
B-Bc
Bx

+B
c
2
A
-2
2
A
Bx

+c
2Bx


B
c
c+
2
A
-2Bx

(1-2
Bx

)
In[]:=
IntegrateLimit
1
A^2*(Exp[-2Bx]-2Exp[-B*x])+2t*x+
Exp[-Bx]
B
+c
,B->0,x(*B=0*)
Out[]=
0
IntegrateLimit
1
A^2*(Exp[-2Bx]-2Exp[-B*x])+2t*x+
Exp[-Bx]
B
+c
,B->Infinity,x(*B->Infinity*)
Out[]=
c+2tx
t
if
2
A
∈&&!(x≤0||c+2tx≤0)
(*Numerically-----------------------------------------------------------------------------------------------------------------------------*)
In[]:=
F[A_,B_,t_,c_]:=NIntegrate
1
A^2*(Exp[-2Bx]-2Exp[-B*x])+2t*x+
Exp[-Bx]
B
+c
,{x,-Infinity,Infinity},Method->"GlobalAdaptive"//Quiet
In[]:=
​​Re[F[1,1/2,-2,2]]
Out[]=
0.203507
In[]:=
Im[F[1,1/2,-2,2]]
Out[]=
-1.921970898101377×
13977
10
(*---------------------------------------------------------------------------------------------------------------------------------------*)
In[]:=
AsymptoticIntegrate
1
A^2*(Exp[-2Bx]-2Exp[-B*x])+2t*x+
Exp[-Bx]
B
+c
,{x,0,1},{B,0,1}
Out[]=
B
t
B
2
t