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Re: Simple integral

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  Re: Simple integral Bob Hanlon <hanlonr@cox.net> - 2011-05-23 12:40 +0000

#2665 — Re: Simple integral

Integrate[1/(x^2 + b x + c), x]

(2*ArcTan[(b + 2*x)/Sqrt[-b^2 + 4*c]])/Sqrt[-b^2 + 4*c]

Simplify[% // TrigToExp, b^2 - 4 c > 0]

(Log[-b + Sqrt[b^2 - 4*c] - 2*x] - Log[b + Sqrt[b^2 - 4*c] + 2*x])/
 Sqrt[b^2 - 4*c]

FullSimplify[%, b^2 - 4 c > 0]

-((2*ArcTanh[(b + 2*x)/Sqrt[b^2 - 4*c]])/Sqrt[b^2 - 4*c])


Bob Hanlon

---- Mariano Pierantozzi <mariano.pierantozzi@gmail.com> wrote: 

=============
Hi,
I've got some problem studing this simple integral:
Integrate[1/(x^2 + b x + c), x].
The Mathematica solution is:
(2 ArcTan[(b + 2 x)/Sqrt[-b^2 + 4 c]])/Sqrt[-b^2 + 4 c]

The problem is that my secon order polinomial have two real solutions, so my
delta (-b^2 + 4 c) is greater than zero. In this case the denominator of the
solution does not exist or exist in complex field, but my x is a volume...
I try in this way
Integrate[1/(x^2 + b x + c), x, Assumptions -> {-b^2 + 4 c < 0}], but
nothing!
I can't trasform my arcotangent in two log.
In summary I would like to have such a solution:
Integrate[1/(x^2 + 5 x + 6), x]
Log[2 + x] - Log[3 + x]
but in general form.
Sorry for my english!
Mariano Pierantozzi
PhD Student
Energy Engineering

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