# Problem with integral of the type (x^2-y^2)^a/2 x^2 Exp[-x]

I tried to calculate the following integral using Mathematica Integrate[(x^2 - y^2)^(3)/2 x^2 Exp[-x], {x, y, Infinity}]

The answer was a conditional expression of the form ConditionalExpression[ 24 E^-y (840 + y (840 + y (375 + y (95 + y (14 + y))))), Re[y] > 0 && Im[y] == 0]

Can I compute the integral with Re[y] less than 0?

## 2 Answers

Or, add the appropriate assumption to Integrate:

Integrate[(x^2 - y^2)^(3)/2 x^2 Exp[-x], {x, y, Infinity},
Assumptions -> y < 0]

(*  24 E^-y (840 + y (840 + y (375 + y (95 + y (14 + y)))))  *)

• Thanks a lot for your feedback – Quantum Fields Dec 12 '20 at 20:38
• I have related question, I tried to compute the following integral Integrate[1/6 Sqrt[x^2 - y^2] x^2 Exp[-x], {x, y, Infinity}] but Mathematica did not give me the answer so how can I compute it ? – Quantum Fields Dec 12 '20 at 20:49
• Your second integral is, of course, complex. If you treat is as a complex line integral along the real axis, the expression would be Integrate[ ... , {x, -y, y, Infinity}, Assumptions -> y > 0] . – LouisB Dec 12 '20 at 22:28

I can not tell you, why there is a conditional expression, but you can do it manually:

In:  i = Integrate[(x^2 - y^2)^(3)/2 x^2 Exp[-x], {x, y, b}];
In:  Limit[i,b->Infinity]
Out: 24 E^-y (840 + 840 y + 375 y^2 + 95 y^3 + 14 y^4 + y^5)