How should I proceed about this integral?
$$\int {1/(x^7 -x)} dx$$
I've tried integration by parts or substitution but I can't seem to solve it. Can I have some hints on how should I get started?
These are some of the things I've tried:
IBP: $u = \frac {1}{x^6-1}$, $du = \frac {-5x^6}{x^6-1}$, $dv = \frac 1x dx$, $v = \ln|x|$
Tried substitution method, but not successful.
Best Answer
One way to look at the problem is to say that it would be easy if the integrand were $$\frac{7x^6-1}{x^7-x},$$and also easy if it were $$\frac{x^6-1}{x^7-x}.$$Now take a linear combination of these to knock out the $x^6$ term in the numerator.