Computing: $ \int \frac{2000x^{2014}+14}{x^{2015}-x}dx $

Question

Computing: $$ \int \frac{2000x^{2014}+14}{x^{2015}-x}dx $$

My work:

$$ I=\int \frac{2000x^{2014}+14}{x^{2015}-x}dx $$

$$ I = \int \frac{2000x^{2014}+14-15+15+15x^{2014}-15x^{2014}}{x^{2015}-x}dx $$

$$ I = \int \frac{2015x^{2014}-1+15-15x^{2014}}{x^{2015}-x}dx $$

$$ I = \int \frac{2015x^{2014}-1}{x^{2015}-x}+\frac{15-15x^{2014}}{x^{2015}-x} dx$$

$$ I = \ln|x^{2015}-x|+ \int \frac{15(1-x^{2014})}{x(x^{2014}-1)}dx$$

$$ I = \ln|x^{2015}-x|+ \int \frac{15(1-x^{2014})}{-x(1-x^{2014})}dx$$

$$ I = \ln|x^{2015}-x| - \int \frac{15}{x} dx$$

$$ I = \ln|x^{2015}-x| - 15\ln|x| + C $$

$$ I = \ln|x^{2015}-x| - \ln|x^{15}| + C $$

$$ I = \ln \left| \frac{x^{2015}-x}{x^{15}} \right| + C $$

Is this correct? If it is please tell if there would be another method of solving this? If it is not please tell me where I went wrong. Thanks :)

Answer 1

Your result is correct, one may also write (simplifying the argument in your last line)

$$ I=\ln \left|x^{2014}-1 \right|-14\ln |x|+C $$

your method is fine to me.