Explanation

(a)
Let $I = \int_{0}^{1}\frac{x^{4}(1 - x)^{4}}{1 + x^{2}}\ dx$
$= \int_{0}^{1}\frac{{(x}^{4} - 1)(1 - x)^{4} + (1 - x)^{4}}{(1 + x^{2})}\ dx$
$= \int_{0}^{1}{\left( x^{2} - 1 \right)(1 - x)^{4}} + \int_{0}^{1}\frac{\left( 1 + x^{2} - 2x \right)^{2}}{(1 + x^{2})}\ dx$
$= \int_{0}^{1}\left\{ \left( x^{2} - 1 \right)(1 - x)^{4} + \left( 1 + x^{2} \right) - 4x + \frac{4x^{2}}{(1 + x^{2})} \right\} dx$
$= \int_{0}^{1}\left( \left( x^{2} - 1 \right)(1 - x)^{4} + \left( 1 + x^{2} \right) - 4x + 4 - \frac{4}{1 + x^{2}} \right)dx$
$= \int_{0}^{1}\left( x^{6} - 4x^{5} + 5x^{4} - 4x^{2} + 4 - \frac{4}{1 + x^{2}} \right)dx$
$= \left\lbrack \frac{x^{7}}{7} - \frac{4x^{6}}{6} + \frac{5x^{5}}{5} - \frac{4x^{3}}{3} + 4x - 4\tan^{- 1}x \right\rbrack_{0}^{1}$
$= \frac{1}{7} - \frac{4}{6} + \frac{5}{5} - \frac{4}{3} + 4 - 4\left( \frac{\pi}{4} - 0 \right)$
$= \frac{22}{7} - \pi$