5.5 The Substitution Rule/11

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

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

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