5.5 The Substitution Rule/5: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
<math> | <math> | ||
\int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} \text{,} \quad u=\cos{(\theta)} | \begin{align} | ||
\int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} \text{,} \quad u=\cos{(\theta)}\\[2ex] | |||
\end{align} | |||
</math> | </math> | ||
<math> | <math> | ||
| Line 12: | Line 16: | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
\int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} &= -\int u^{3}du \\[2ex] | \int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} &= -\int u^{3}du \\[2ex] | ||
&= \frac{-u^{4}}{4} + C = \frac{-\cos^{4}{(\theta)}}{4} + C | &= \frac{-u^{4}}{4} + C = \frac{-\cos^{4}{(\theta)}}{4} + C \\[2ex] | ||
&= \frac{-1}{4}\cos^{4}{(\theta)} + C | &= \frac{-1}{4}\cos^{4}{(\theta)} + C | ||
Latest revision as of 19:52, 22 September 2022
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}{\text{,}}\quad u=\cos {(\theta )}\\[2ex]\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=\cos {(\theta )}\\[2ex]du&=-\sin {(\theta )}d{(\theta )}\\[2ex]-du&=\sin {(\theta )}d{(\theta )}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e43c3b8a3210b14665689ce0e483eaeaf5313e33)
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}&=-\int u^{3}du\\[2ex]&={\frac {-u^{4}}{4}}+C={\frac {-\cos ^{4}{(\theta )}}{4}}+C\\[2ex]&={\frac {-1}{4}}\cos ^{4}{(\theta )}+C\end{aligned}}}