5.5 The Substitution Rule/27: Difference between revisions
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<math> | |||
\int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz | |||
</math> | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
& \ | u &=1+{z}^3 \\[2ex] | ||
du &=3{z}^2dz \\[2ex] | |||
\frac{1}{3}du &={z}^2dz \\[2ex] | |||
\end{align} | |||
</math> | |||
<math> | |||
\begin{align} | |||
\int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz &= \frac{1}{3}\int\frac{1}{\sqrt[3]{u}}du = \frac{1}{3}\int{u}^{-\frac{1}{3}}du \\[2ex] | |||
&= -\frac{1}{3}(\frac{3}{2}{u}^\frac{2}{3}) = \frac{3}{6}{u}^{2/3} \\[2ex] | |||
&= \frac{1}{2}({1+z^{3}})^\frac{2}{3} + C | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 16:38, 7 September 2022
![{\displaystyle \int {\cfrac {z^{2}}{\sqrt[{3}]{1+z^{3}}}}dz}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a16762723c154c008294ac839f1997ba1455d543)
![{\displaystyle {\begin{aligned}u&=1+{z}^{3}\\[2ex]du&=3{z}^{2}dz\\[2ex]{\frac {1}{3}}du&={z}^{2}dz\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a2fa06923c4c79966cfebb815eb85a2321de6418)
![{\displaystyle {\begin{aligned}\int {\cfrac {z^{2}}{\sqrt[{3}]{1+z^{3}}}}dz&={\frac {1}{3}}\int {\frac {1}{\sqrt[{3}]{u}}}du={\frac {1}{3}}\int {u}^{-{\frac {1}{3}}}du\\[2ex]&=-{\frac {1}{3}}({\frac {3}{2}}{u}^{\frac {2}{3}})={\frac {3}{6}}{u}^{2/3}\\[2ex]&={\frac {1}{2}}({1+z^{3}})^{\frac {2}{3}}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8cf3ab6276fae0f2c5fb9ea43317348c08d2fb57)