(dx)/(x^7-x)

Question

find the integral of (dx)/(x^7-x)

Answer 1

∫1/(x^7-x) dx

 

1/(x^7-x) ==>> 1/((1-1/x^6) x^7)

= ∫ 1/((1-1/x^6) x^7) dx

 

substitute u = 1-1/x^6  and  du = 6/x^7 dx

 

= 1/6 ∫ 1/u du

The integral of 1/u is ln(u)

 

= (ln(u))/6+constant

Substitute back for u = 1-1/x^6

 

  = 1/6 ln(1-1/x^6)+constant

Which is equivalent for restricted x values to:

  = 1/6 ln(1-x^6)-ln(x)+constant