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### Re: Converting the integral of a sum into a sum of integrals

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```I don't know why you have patterns in the rhs of the rule. In any case, the
following works. I set the a_i coefficients to 1 in the last step to obtain
a complete evaluation.

HoldForm@Integrate[
Sum[Times[Power[x, i], Subscript[a, i]], {i, 0, k}], x]
% /. Integrate[Sum[term_, range_], variable_] :>
Sum[Integrate[term, variable], range]
% // ReleaseHold
% /. Subscript[a, i] -> 1

David Park
djmpark@comcast.net
http://home.comcast.net/~djmpark/

From: Kurt TeKolste [mailto:tekolste@fastmail.net]

If I ask Mathematica to perform an indefinite integrate on a symbolic
summation (with terms of the form a_i x^i),

Integrate[Sum[Times[Power[x,i],Subscript[a,i]],{i,0,k}],

it does nothing.  If I try to tell it that summation and integration
commute by applying the rule:

Integrate[Sum[term_,range_],variable_]->
Sum[Integrate[term_,variable_],range_]

I get a strange result equivalent to

term * range

or, in this case,

Times[Power[x,i],Subscript[a,i]] * {i,0,k}
or
{i a_i x^i, 0, k a_i x^i}

Any ideas as to what's going on?

(BTW:

1)  If you use rules to extract each of term, range, and variable and
then take Sum[Integral[...],...] the correct answer is returned.

2)  if you use a rule that changes Sum to Power, i.e.

Integrate[Sum[term_,range_],variable_] ->
Product[Integrate[term_,variable_],range_]

you get exponents

term^(range - 1)

Odd...
)

ekt

```
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