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