Explanation

(a)
Let $I = \int_{}^{}{(\sin^{6}{x + \cos^{6}{x + 3\sin^{2}{x\cos^{2}{x)dx}}}}}$
$= \int_{}^{}{\{({\sin^{2}{x)}}^{3} + ({\cos^{2}{x)}}^{3} + 3\sin^{2}{x\cos^{2}{x\}}dx}}$
$= \int_{}^{}\left\lbrack \begin{array}{r}
{(sin}^{2}{x + \cos^{2}{x)(\sin^{4}{x + \cos^{4}x}}} \\
- \sin^{2}{x\cos^{2}{x) + 3\sin^{2}{x\cos^{2}x}}}
\end{array} \right\rbrack dx$
$= \int_{}^{}\left\lbrack \begin{array}{r}
(\sin^{2}{x + \cos^{2}{x) - 3\sin^{2}{x\cos^{2}x}}} \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 3\sin^{2}{x\cos^{2}x}
\end{array} \right\rbrack dx$
$= \int_{}^{}1\ dx = x + c$