#1**+1 **

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**

Guest May 17, 2017