5.5 The Substitution Rule/65

$\int _{1}^{2}x{\sqrt {x-1}}dx$

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{1}^{2}(x{\sqrt {x-1}}\,)\;dx&=\int _{0}^{1}((u+1){\sqrt {u}})\;du=\int _{0}^{1}(u^{\frac {3}{2}}+{\sqrt {u}})\;du\\[2ex]&=({\frac {2}{5}}u^{\frac {5}{2}}+{\frac {2}{3}}u^{\frac {3}{2}}){\bigg |}_{0}^{1}={\frac {2}{5}}+{\frac {2}{3}}\\[2ex]&={\frac {16}{15}}\\[2ex]\end{aligned}}}