Integrating with assumptions and limits of integration

Question

I have an integration with assumptions and limits of integration. I tried to use Mathematica to solve this problem, but I cannot get any results.

Integrate[1/(w^4 + 2 (2 ξ^2 - 1) w^2 ω1^2 + ω1^4), {ω1, 0, t}, 
  Assumptions -> {w > 0, 1 > ξ > 0}]]

What should I do?

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This could be considered a duplicate of How to do symbolic definite integral without copy and paste the intermediate results? — Jens, Mar 07 '15 at 04:54
I agree with @Jens. All those definite integrals that can be solved by doing the indefinite one should be "the same" question, lest we are going to repeat the trick over and over — Dr. belisarius, Mar 07 '15 at 15:14

bbgodfrey · Answer 1 · 2015-03-07T02:17:11.667

2

Evaluating the indefinite integral works better.

sol = Integrate[1/(w^4 + 2 (2 v^2 - 1) w^2 w1^2 + w1^4), w1, Assumptions -> {w > 0, 1 > v > 0}]

(* (-(ArcTan[w1/(Sqrt[-1 + 2*v^2 + 2*v*Sqrt[-1 + v^2]]*w)]/
   Sqrt[-1 + 2*v^2 + 2*v*Sqrt[-1 + v^2]]) - 
   ArcTanh[w1/(Sqrt[1 - 2*v^2 + 2*v*Sqrt[-1 + v^2]]*w)]/
   Sqrt[1 - 2*v^2 + 2*v*Sqrt[-1 + v^2]])/(4*v*Sqrt[-1 + v^2]*w^3) *)

If the limits of integration are important, then evaluate

sol/.w1 -> t - sol/.w1 -> 0

(The second term, incidently, is 0.)

edited Mar 07 '15 at 02:17

answered Mar 07 '15 at 01:46

bbgodfrey

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@Nasser I don't see anything wrong with this answer. Just different letters. – Jens Mar 07 '15 at 04:53
@Jens oh sorry, I saw 2 w's in there. did not notice one had 1 stuck to it. – Nasser Mar 07 '15 at 05:10
Actually I want the final limits, w1->infinity-w1->0, and the assumption 1>v>0.@bbgodfrey – whdwpy666 Mar 09 '15 at 01:46
@whdwpy666 The limits you request were not part of your Question but can applied easily to obtain, (Pi*(Sqrt[-(1/((1 - 2*v^2 + 2*v*Sqrt[-1 + v^2])*w^2))] - Sqrt[1/((-1 + 2*v^2 + 2*v*Sqrt[-1 + v^2])*w^2)]))/(8*v*Sqrt[-1 + v^2]*w^2) The assumption 1>v>0 was used to obtain the answer, as requested. – bbgodfrey Mar 09 '15 at 02:16

Integrating with assumptions and limits of integration

1 Answers1