5.5 The Substitution Rule/11

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

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} u &=2x+x^{2} \\[2ex] du &=2+2x dx \\[2ex] \frac{1}{2}du &=x+1 \end{align} }

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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}}}