# Question:constant of integration

## Question:constant of integration

Maple

Dear friends,

I have a basic question concerning Maple 15 (X86 64 LINUX). In fact it is so simple it is almost embarassing! Here it is.

When we do integration in a basic calculus class we learn that when integrating polynomials and series the constant that appears during the integration is usually taken to be zero, as polynomial and Laurent series terms will only produce powers and logs, but no constant. Of course the choice of constant does not influence definite integrals, as it cancels.

Now when we try this e.g. with

`int(x^3/(1+x)^2/(1+x^2), x)`

and look at the series expansion of the result around zero we get

`1/2 + 1/4 x^4  - 2/5 x^5  + O(x^6)`

we see that the constant that appeared during the integration is actually

`1/2.`

This could not have appeared during the integration since the integrand has series

`x^3  - 2 x^4  + 2 x^5  + O(x^6),`

none of whose terms will produce a constant.

My question is, how does Maple determine this choice of constant? Is there a formula for the constant other than substituting zero into the integration result? In this particular example we have a term in

`1/2 arctan`

which is multivalued, so that the constant could just as well be

`π + 1/2.`

Somehow Maple is choosing constants so that everything fits together, but how? I suspect this has to do with the choice of branch of the logarithm. BTW it would be nice if Maple were to offer a built in logarithm where we can specify the range of the argument that it returns. That would save a certain amount of programming effort.

Thanks ever so much!

Marko Riedel

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