Explanation

(c)
Let $I = \int_{}^{}\frac{dx}{x(x^{5} + 1)}$
$= \int_{}^{}\frac{x^{4}dx}{x^{5}(x^{5} + 1)}$
Put $x^{5} = t$ and $x^{4}dx = \frac{1}{5}\ dt$
$\therefore\ I = \frac{1}{5}\ \int_{}^{}\frac{dt}{t(t + 1)}$
$= \frac{1}{5}\int_{}^{}\left\lbrack \frac{1}{t} - \frac{1}{t + 1} \right\rbrack dt$
$= \frac{1}{5}\left\lbrack \log t - \log(t + 1) \right\rbrack + c$
$= \frac{1}{5}\log\frac{t}{t + 1} + c$
$= \frac{1}{5}\log\frac{x^{5}}{x^{5} + 1} + c$