Explanation

(b)
Let $I = \int_{}^{}{32x^{3}}\left( \log x \right)^{2}\ dx$
$= 32\ \left\{ \left( \log x \right)^{2}\frac{x^{4}}{4} - \int_{}^{}{2\log x.\frac{1}{x}.\frac{x^{4}}{4}}dx \right\}$
$= 8x^{4}\left( \log x \right)^{2} - 16\left\{ \log{x.}\frac{x^{4}}{4} - \int_{}^{}{\frac{1}{x}.\frac{x^{4}}{4}dx} \right\}$
$= 8x^{4}\left( \log x \right)^{2} - 4x^{4}\log{x + x^{4}} + c$
$= x^{4}\left\lbrack 8\ \left( \log x \right)^{2} - 4\log{x + 1} \right\rbrack + c$