5.3 The Fundamental Theorem of Calculus/33

${\begin{aligned}\int _{1}^{2}\left(1+2y\right)^{2}dy&=\int _{1}^{2}\left(1+4y+4y^{2}\right)dy\\[2ex]&=y+{\frac {4y^{2}}{2}}+{\frac {4y^{3}}{3}}{\bigg |}_{1}^{2}\\[2ex]&=\left[2+{\frac {4(2)^{2}}{2}}+{\frac {4(2)^{3}}{2}}\right]-\left[1+{\frac {4(1)^{2}}{2}}+{\frac {4(1)^{3}}{2}}\right]\\[2ex]&={\frac {49}{3}}\end{aligned}}$