IMathAS-libretexts

Evaluate using the substitution method.

$\displaystyle \int\frac{{{2}{x}^{{7}}}}{{\left({4}+{x}^{{8}}\right)}^{{7}}}{\left.{d}{x}\right.}$

$\displaystyle \int{\left(\frac{{{3}{u}^{{-{7}}}}}{{8}}{d}{u}\right)}={0}\cdot{\left({4}+{x}^{{8}}\right)}^{{-{{6}}}}+{C}$
$\displaystyle \int{\left(\frac{{{3}{u}^{{-{7}}}}}{{8}}{d}{u}\right)}=-\frac{{1}}{{24}}\cdot{\left({4}+{x}^{{8}}\right)}^{{7}}+{C}$
$\displaystyle \int{\left(\frac{{{u}^{{{7}}}-{u}}}{{28}}{d}{u}\right)}=\frac{{\frac{{{2}{x}^{{8}}}}{{8}}}}{{\frac{{1}}{{8}}{\left({4}{x}+\frac{{x}^{{8}}}{{8}}\right)}^{{8}}}}+{C}$
$\displaystyle \int{\left(\frac{{{u}^{{-{7}}}}}{{4}}{d}{u}\right)}=-\frac{{1}}{{24}}\cdot{\left({4}+{x}^{{8}}\right)}^{{-{{6}}}}+{C}$
$\displaystyle \int{\left({2}{u}^{{{7}}}{d}{u}\right)}=-\frac{{1}}{{24}}\cdot\frac{{1}}{{\left({4}+{x}^{{8}}\right)}^{{6}}}+{C}$