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Re: Why does this lead to an answer with complex numbers?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71472] Re: Why does this lead to an answer with complex numbers?
*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
*Date*: Mon, 20 Nov 2006 02:43:54 -0500 (EST)
*Organization*: The Open University, Milton Keynes, UK
*References*: <ejosmm$n3k$1@smc.vnet.net>
aaronfude at gmail.com wrote:
> The expression is
>
> \!\(FullSimplify[
> Assuming[\[Beta] > 0 && \[Beta] < Pi/2,
> Integrate[\(-Log[\@\(1 + x\^2\) - 1/11*x\ ]\), \ x]]]\)
Two remarks: the integral is complex and it is independent of any
variable or constant called beta.
FullSimplify[Assuming[\[Beta] > 0 && \[Beta] < Pi/2,
Integrate[-Log[Sqrt[1 + x^2] - (1/11)*x], x]]]
returns
x 2
x - x Log[-(--) + Sqrt[1 + x ]] -
11
1 2 Sqrt[30] x
---------- (11 (4 ArcTan[------------] +
8 Sqrt[30] 11
2
4 ArcTan[2 Sqrt[30] Sqrt[1 + x ]] +
2 2
I (2 Log[900 (121 + 120 x ) ] -
2
Log[(121 + 120 x )
2 2
(-121 - 122 x + 22 x Sqrt[1 + x ])] -
2
Log[(121 + 120 x )
2 2
(121 + 122 x + 22 x Sqrt[1 + x ])])))
Integrate[-Log[Sqrt[1 + x^2] - (1/11)*x], x]
yields
1 2 Sqrt[30] x
--- (240 x - 44 Sqrt[30] ArcTan[------------] -
240 11
2
44 Sqrt[30] ArcTan[2 Sqrt[30] Sqrt[1 + x ]] -
2 2
22 I Sqrt[30] Log[900 (121 + 120 x ) ] -
x 2
240 x Log[-(--) + Sqrt[1 + x ]] +
11
2
11 I Sqrt[30] Log[(121 + 120 x )
2 2
(-121 - 122 x + 22 x Sqrt[1 + x ])] +
2
11 I Sqrt[30] Log[(121 + 120 x )
2 2
(121 + 122 x + 22 x Sqrt[1 + x ])])
FreeQ[%, \[Beta]]
--> True
Regards,
Jean-Marc
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