Re: Simple integral
- To: mathgroup at smc.vnet.net
- Subject: [mg119192] Re: Simple integral
- From: Stefan Salanski <wutchamacallit27 at gmail.com>
- Date: Tue, 24 May 2011 05:59:21 -0400 (EDT)
- References: <irdctv$3tq$1@smc.vnet.net>
On May 23, 6:29 am, Mariano Pierantozzi <mariano.pieranto... at 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 Mariano, It turns out that since the imaginary quantity Sqrt[-b^2+4c] appears in both the denominator, AND in ArcTan in your numerator, they cancel out and you are left with a real answer whenever you actually evaluate it, so plugging in numbers b and c you will get real answers all the time (unless your bounds of integration include either of the two poles of this equation, then you get errors because it blows up). The relation is between arctan and arctanh, the hyperbolic tangent inverse. ArcTan[ x * I ] = I * ArcTanh[x] what we can do to simplify your result is to use this relation to cancel the I's manually, i get -(ArcTanh[(b + 2 x)/Sqrt[b^2 - 4 c]]/(2 Sqrt[b^2 - 4 c])) then TrigToExp gives Log[1 - (b + 2 x)/Sqrt[b^2 - 4 c]]/(4 Sqrt[b^2 - 4 c]) - Log[1 + (b + 2 x)/Sqrt[b^2 - 4 c]]/(4 Sqrt[b^2 - 4 c]) which I believe is the general case. -Stefan S