# Integrating with assumptions and limits of integration [duplicate]

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 7, 2015 at 4:54
• 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 Mar 7, 2015 at 15:14

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.)

• @Nasser I don't see anything wrong with this answer. Just different letters.
– Jens
Mar 7, 2015 at 4:53
• @Jens oh sorry, I saw 2 w's in there. did not notice one had 1 stuck to it. Mar 7, 2015 at 5:10
• Actually I want the final limits, w1->infinity-w1->0, and the assumption 1>v>0.@bbgodfrey Mar 9, 2015 at 1:46
• @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. Mar 9, 2015 at 2:16