5.5 The Substitution Rule/30

$\int {\frac {\sin {(\ln {(x))}}}{x}}dx$

${\begin{aligned}u&=2+x^{4}\\[2ex]du&=4x^{3}dx\\[2ex]{\frac {1}{4}}du&=x^{3}dx\end{aligned}}$

${\begin{aligned}\int x^{3}(2+x^{4})^{5}dx&=\int (x^{3}dx)(2+x^{4})=\int \left({\frac {1}{4}}du\right)(u)={\frac {1}{4}}\int u\,du\\[2ex]&={\frac {1}{4}}\left[{\frac {u^{1+1}}{1+1}}\right]+C={\frac {u^{2}}{8}}+C\\[2ex]&={\frac {(2+x^{4})^{2}}{8}}+C\end{aligned}}$