5.4 Indefinite Integrals and the Net Change Theorem/30

${\begin{aligned}\int _{1}^{2}{\frac {y+5y^{7}}{y^{3}}}dy&=\int _{1}^{2}\left({\frac {y}{y^{3}}}+{\frac {5y^{7}}{y^{3}}}\right)dy=\int _{1}^{2}(y^{-2}+5y^{4})dy\\[2ex]&=\left({\frac {y^{-2+1}}{-2+1}}+{\frac {5y^{4+1}}{4+1}}\right){\bigg |}_{1}^{2}=\left({\frac {y^{-1}}{-1}}+y^{5}\right){\bigg |}_{1}^{2}=\left(-{\frac {1}{y}}+y^{5}\right){\bigg |}_{1}^{2}\\[2ex]&=\left(-{\frac {1}{(2)}}+(2)^{5}\right)-\left(-{\frac {1}{(1)}}+(1)^{5}\right)\\[2ex]&=\left(-{\frac {1}{2}}+32\right)=\left(-{\frac {1}{2}}+{\frac {64}{2}}\right)={\frac {63}{2}}\end{aligned}}$