Explanation

(b)
Let $I = \int_{}^{}\frac{dx}{x(x^{7} + 1)}$
Put $x^{7} = t\ \ \Rightarrow \ dx = \frac{1}{7x^{6}}dt$
$\therefore\ I = \frac{1}{7}\int_{}^{}\frac{dt}{t(t + 1)} = \frac{1}{7}\int_{}^{}\left( \frac{1}{t} - \frac{1}{t + 1} \right)dt$
$= \frac{1}{7}\left\lbrack \log{t - \log(t + 1)} \right\rbrack + c$
$= \frac{1}{7}\log{\left( \frac{t}{t + 1} \right) + c}$
$= \frac{1}{7}\log\left( \frac{x^{7}}{x^{7} + 1} \right) + c$