5.5 The Substitution Rule/11

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

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

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