• 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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