5.4 Indefinite Integrals and the Net Change Theorem/27

${\begin{aligned}\int _{1}^{4}{\sqrt {t}}(1+t)dt&=\int _{1}^{4}\left(t^{\frac {1}{2}}+t^{\frac {3}{2}}\right)dt\\[2ex]&=\left({\frac {2(t)^{3/2}}{3}}+{\frac {2(t)^{5/2}}{5}}\right)|_{1}^{4}\\[2ex]&={\frac {2(t)^{3/2}}{3}}+{\frac {2(t)^{5/2}}{5}}{\bigg |}_{1}^{4}&={\frac {2(4)^{3/2}}{3}}+{\frac {2(4)^{5/2}}{5}}-{\frac {2(1)^{3/2}}{3}}+{\frac {2(1)^{5/2}}{5}}&={\frac {256}{15}}\end{aligned}}$