>>11539322Let e(y) be shorthand for e^(2*pi*i*y)
1/(1+x^5) = 1 - x^5 + x^10 - x^15 ...
Integrating gives x - x^6/6 + x^11/11 - x^16/16 ... + C
How can we construct this from simpler functions?
log(1+x) = x - x^2/2 + x^3/3 ...
Let f(x) = log(1+x)/x = 1 - x/2 + x^2/3 - x^3/4 ....
The answer is:
(x/5)*[f(x*e(0/5)) + f(x*e(1/5)) + f(x*e(2/5)) + f(x*e(3/5)) + f(x*e(4/5))] + C
Summing over the fifth roots of unity like that kills all powers that aren't a multiple of 5 and multiplies all powers that are multiples of 5 by a factor of 5.
I divided log(1+x) by x to shift the /1, /6, /11, /16. etc. down to powers that are multiples of 5.
multiplying by (x/5) shifts them back up and takes care of the extra factor of 5.
I don't have time for partial fraction bs.
