5.5 The Substitution Rule/9: Difference between revisions
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<math> | <math> | ||
\int (3x-2)^20 dx | \int (3x-2)^{20} dx | ||
</math> | </math> | ||
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\begin{align} | \begin{align} | ||
u &= | u &=3x-2 \\[2ex] | ||
du &= \frac{1}{ | du &= 3dx \\[2ex] | ||
\frac{1}{3}du &= dx \\[2ex] | |||
\end{align} | \end{align} | ||
</math> | |||
<math> | |||
\begin{align} | |||
\int u^{20}\cdot\frac{1}{3}du \\[2ex] | |||
&= \frac{1}{3}\int u^{20}du | |||
&= \frac{1}{3}\cdot\frac{1}{20+1} u^{20+1} | |||
&= \frac{1}{3}\cdot\frac{1}{21} u^{21} | |||
&= \frac{1}{63} u^{21} + c | |||
&= \frac{1}{63} (3x-2)^{21} + c | |||
\end{align} | |||
</math> | |||
<math> | |||
=\frac{1}{63} (3x^21-2097152) + c | |||
</math> | </math> | ||
Latest revision as of 19:37, 20 September 2022
![{\displaystyle {\begin{aligned}u&=3x-2\\[2ex]du&=3dx\\[2ex]{\frac {1}{3}}du&=dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/45bebb73a627f87f28c7b799f9d7ce3e8d4e6fd5)
![{\displaystyle {\begin{aligned}\int u^{20}\cdot {\frac {1}{3}}du\\[2ex]&={\frac {1}{3}}\int u^{20}du&={\frac {1}{3}}\cdot {\frac {1}{20+1}}u^{20+1}&={\frac {1}{3}}\cdot {\frac {1}{21}}u^{21}&={\frac {1}{63}}u^{21}+c&={\frac {1}{63}}(3x-2)^{21}+c\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/504d7bdccf3dc2ef9663dd875868f1329b9c7594)
