Question

§(1-x^7)/(x(1+x^7))dx

Answer 1

I=∫(1-x^7)/(x(1+x^7))dx ----(1)

=∫1/x - 2∫x^7/(x(1+x^7)) dx
= lnx + 2(I-∫1/(x(1+x^7)) dx
I = -lnx + 2∫1/(x(1+x^7)) dx ----(2)

2*(1) - (2)
I = lnx - 2∫x^7/(x(1+x^7))dx
= lnx - 2∫ x^6/(1+x^7) dx

let y = 1+x^7
dy = 7x^6dx
∫ x^6/(1+x^7) dx
= 1/7∫ 1/y dy
=1/7 lny
=1/7 ln(1+x^7)

I = lnx - 2∫ x^6/(1+x^7) dx
= lnx - 2/7 ln(1+x^7) + C #

Answer 2

∫(1-x^7)/x(1+x^7)dx

=∫[1/x - 2x^6/(1+x^7)] dx

= ln|x| - 2/7∫[1/(1+x^7)] d[1+x^7)

= ln|x| -2/7*ln|1+x^7| + C