Mathematica Asked by Dimitri_896 on April 25, 2021
So I have this function:
$$ int_{-infty}^{infty} mathrm {sech}(x+s)^{2} {rm sech}(x)^{2}dx$$
And when I try to integrate it, I can obtain the indefinite integral:
In[42]:= Integrate[Cosh[x + s]^-2*Cosh[x]^-2, x]
Out[42]= -2 Coth[s] Csch[s]^2 Log[Cosh[x]]+2Coth[s] Csch[s]^2 Log[Cosh[s + x]]-Csch[s]^2 Sech[s] Sech[s+x] Sinh[x]-Csch[s]^2Tanh[x]
But when I evaluate the limits, it cancels to $0$. The solution I was given stated that the answer should be some form of:
$$frac{cosh(s)cdot s}{sinh(s)^3}- frac{1}{sinh(s)^2} $$
None of what I’m doing seems to get me the answer and as you can see, it’s not like Mathematica even makes the output easy to parse.
Try the definite integral
Integrate[Cosh[x + s]^-2*Cosh[x]^-2, {x, -Infinity,Infinity}]
(*2 Csch[s]^2 (-2 + Coth[s] Log[E^(2 s)]) if ...*)
which Mathematica conditional solves.
Numerical "confirmation"
int[s_?NumericQ] :=
NIntegrate[
Cosh[x + s]^-2*Cosh[x]^-2,
{x, -Infinity, Infinity}]
Plot[{int[s], 2 Csch[s]^2 (-2 + Coth[s] Log[E^(2s)])} , {s, -10, 10},PlotStyle -> {Automatic, Dashed}]
Answered by Ulrich Neumann on April 25, 2021
$Version
(* "12.2.0 for Mac OS X x86 (64-bit) (December 12, 2020)" *)
Clear["Global`*"]
expr = Cosh[x + s]^-2*Cosh[x]^-2;
An indefinite integral (anti-derivative) of expr
is
ad = Integrate[expr, x] // Simplify
(* -Csch[s]^2 (2 Coth[s] (Log[Cosh[x]] - Log[Cosh[s + x]]) +
Sech[s] Sech[s + x] Sinh[x] + Tanh[x]) *)
Verifying that ad
is a valid anti-derivative of expr
D[ad, x] == expr // Simplify
(* True *)
The definite integral is then
int1 = Limit[ad, x -> Infinity] - Limit[ad, x -> -Infinity] // Simplify
(* -2 Csch[s]^2 (2 + Coth[s] (Log[E^-s] - Log[E^s])) *)
Calculating the definite integral directly
int2 = Integrate[expr, {x, -Infinity, Infinity}]
The results are equivalent for real s
diff = int1 - int2 // Simplify[#, Element[s, Reals]] &
(* 0 *)
Graphically,
Plot[{int1, int2}, {s, -10, 10},
PlotStyle -> {Automatic, Dashed}]
Answered by Bob Hanlon on April 25, 2021
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