5.5 The Substitution Rule/11: Difference between revisions
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\begin{align} | \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] | \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{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 21:28, 22 September 2022
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 (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] &= \frac{1}{3}(2x+x^{2})^{\frac{3}{2}} + C \\[2ex] \end{align} }