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I just spent some time tracking down a strange problem in my code. I
wonder if anyone has seen something like this. It seems that
Integrate[] gets  confused about which variable it is integrating
over, when you give it a series to integrate:

In[827]:= Integrate[Log[a]+2q a x^2+3w a^2x+O[a]^4+x,x]
Integrate[Log[a]+2q a x^2+3w a^2x+O[a]^4+x,a]
Out[827]= (-1+x+Log[a]) a+q x^2 a^2+w x a^3+O[a]^5
Out[828]= (-1+x+Log[a]) a+q x^2 a^2+w x a^3+O[a]^5

It seems that in both cases Integrate actually used a as the variable,
even though I asked for x in the first case. I see this with version
5.2 and 6.0.

Without the Log[a] term it all works as expected:
In[829]:= Integrate[2q a x^2+3w a^2x+O[a]^4+x,x]
Integrate[2q a x^2+3w a^2x+O[a]^4+x,a]
Out[829]= x^2/2+2/3 q x^3 a+3/2 w x^2 a^2+O[a]^4
Out[830]= x a+q x^2 a^2+w x a^3+O[a]^5

Or am I misunderstanding the use of O[] in mathematica, or is this a
bug? After all, the strict mathematical interpretation of O[] as big-O
notation has that "O(x^4)  is f(x) + O(x^5) as x->inf"
from which point of view, mathematica is strictly correct there, just
not giving the tightest bound it could, and throwing some misleading
terms in as well.

On the other hand, I don't get the idea that the O[n] is supposed to
be interpreted as mathematical 'big-O' O(n), but rather as the lowest
order missing term in a truncated Taylor/Laurent/Puiseux/etc series


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