5.4 Indefinite Integrals and the Net Change Theorem/27

${\begin{aligned}\int _{1}^{4}{\sqrt {t}}(1+t)dt&=\int _{1}^{4}\left(t^{\frac {1}{2}}+t^{\frac {3}{2}}\right)dt\\[2ex]&=\left({\frac {2t^{\frac {3}{2}}}{3}}+{\frac {2t^{\frac {5}{2}}}{5}}\right){\Bigg |}_{1}^{4}\\[2ex]&=\left[{\frac {2(4)^{\frac {3}{2}}}{3}}+{\frac {2(4)^{\frac {5}{2}}}{5}}\right]-\left[{\frac {2(1)^{\frac {3}{2}}}{3}}+{\frac {2(1)^{\frac {5}{2}}}{5}}\right]\\[2ex]&=\left[{\frac {16}{3}}+{\frac {64}{5}}\right]-\left[{\frac {2}{3}}+{\frac {2}{5}}\right]\\[2ex]&={\frac {256}{15}}\end{aligned}}$