5.5 The Substitution Rule/65

$\int _{1}^{2}x{\sqrt {x-1}}dx=\int _{0}^{1}u+1{\sqrt {u}}du=\int _{0}^{1}(u+1)({\sqrt {u}})=\int _{0}^{1}u^{\frac {3}{2}}+{\sqrt {u}}du={\frac {2}{5}}U^{\frac {5}{2}}+{\frac {2}{3}}U^{\frac {3}{2}}|_{0}^{1}={\frac {2}{5}}+{\frac {2}{3}}={\frac {16}{15}}$

${\begin{aligned}u&=x-1\\u+1&=x\\[2ex]du&=2dx\\[2ex]{\frac {1}{2}}du&=dx\end{aligned}}$

<math> \begin{align} \int_{1}^{2} x \sqrt{x-1} dx = \int_{0}^{1} u+1 \sqrt{u}du = \int_{0}^{1}(u + 1)(\sqrt{u}) = \int_{0}^{1} u^ \frac{3}{2} + \sqrt{u}du = \frac{2}{5} U^\frac{5}{2} + \frac{2}{3} U^\frac{3}{2}| _{0}^{1} =\frac{2}{5} + \frac{2}{3} = \frac{16}{15}