Explanation

(d)
Let $I = \int_{}^{}{\frac{mx^{m + 2n - 1} - nx^{n - 1}}{x^{2m + 2n} + 2x^{m + n} + 1}\ dx}$
$= \int_{}^{}{\frac{mx^{m + 2n - 1} - nx^{n - 1}}{\left( 1 + x^{m + n} \right)^{2}}dx}$
$\therefore I = \int_{}^{}{\frac{mx^{m - 1} - \frac{n}{x^{n + 1}}}{\left( \ x^{m} + \frac{1}{x^{n}}\ \right)^{2}}dx}$
Let $x^{m} + \frac{1}{x^{n}} = t$
$\Rightarrow mx^{m - 1} - \frac{n}{x^{n + 1}}dx = dt$
$\therefore I = \int_{}^{}{\frac{1}{t^{2}}dt = - \frac{1}{t} + c}$
$= \frac{- 1}{\left( \ x^{m} + \frac{1}{x^{n}}\ \right)} + c = \frac{- x^{n}}{x^{m + n} + 1} + c$