Explanation

(a)
Let $I = \int_{}^{}\frac{(1 + x^{4})}{\left( 1 - x^{4} \right)^{\frac{3}{2}}}\ dx$
$= \int_{}^{}\frac{x^{2}\left( \frac{1}{x^{2}} + x^{2} \right)}{x^{3}\left( \frac{1}{x^{2}} - x^{2} \right)}dx = \int_{}^{}\frac{\left( \frac{1}{x^{3}} + x \right)}{\left( \frac{1}{x^{2}} - x^{2} \right)}\ dx$
Put $\frac{1}{x^{2}} - x^{2} = z$
$\Rightarrow \left( \frac{1}{x^{3}} + x \right)dx = - \frac{dz}{z}$
$\therefore I = - \frac{1}{2}\int_{}^{}\frac{dz}{z^{\frac{3}{2}}}$
$= z^{\frac{- 1}{2}} + c = \frac{x}{\sqrt{1 - x^{4}}} + c$