Explanation

(c)
$\int_{}^{}{\frac{x^{2} - 1}{x^{4} + x^{2} + 1}\ dx}$
$\Rightarrow I = \int_{}^{}{\frac{1 - \frac{1}{x^{2}}}{x^{2} + \frac{1}{x^{2}} + 1}dx}$
$= \int_{}^{}{\frac{1 - \frac{1}{x^{2}}}{\left( \ x - \frac{1}{x}\ \right)^{2} - 1}dx}$
Put $x + \frac{1}{x} = t\ \Rightarrow \left( \ 1 - \frac{1}{x^{2}}\ \right)dx = dt$
$\therefore I = \int_{}^{}{\frac{dt}{t^{2} - 1^{2}} = \frac{1}{2 \times 1}\log{\frac{t - 1}{t + 1} + c}}$
$= \frac{1}{2}\log{\frac{x^{2} + 1 - x}{x^{2} + 1 + x} + c}$