5.5 The Substitution Rule/11

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

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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] \end{align} }