Take the integral:
integral(2000 x^2014 + 14)/(x^2015 - x) dx
For the integral (2000 x^2014 + 14)/(x^2015 - x), write (2000 x^2014 + 14)/(x^2015 - x) as (2000 x^1999 + 14/x^15)/(x^2000 - 1/x^14):
= integral(2000 x^1999 + 14/x^15)/(x^2000 - 1/x^14) dx
For the integrand (2000 x^1999 + 14/x^15)/(x^2000 - 1/x^14), substitute u = x^2000 - 1/x^14 and du = (2000 x^1999 + 14/x^15) dx:
= integral1/u du
The integral of 1/u is log(u):
= log(u) + constant
Substitute back for u = x^2000 - 1/x^14:
= log((x^2014 - 1)/x^14) + constant
Which is equivalent for restricted x values to:
Answer: | = log(1 - x^2014) - 14 log(x) + constant
