5.5 The Substitution Rule/11

$\int (x+1){\sqrt {2x+x^{2}}}dx$

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

${\begin{aligned}\int (x+1){\sqrt {2x+x^{2}}}dx&={\frac {1}{2}}\int {\sqrt {u}}du={\frac {1}{2}}\int u^{\frac {1}{2}}du\\[2ex]&={\frac {1}{2}}({\frac {2u^{\frac {3}{2}}}{3}})+C\\[2ex]&={\frac {1}{3}}(u^{\frac {3}{2}})+C\\[2ex]&={\frac {1}{3}}(2x+x^{2})^{\frac {3}{2}}+C\\[2ex]\end{aligned}}$