5.5 The Substitution Rule/5: Difference between revisions
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<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> | ||
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\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
![{\displaystyle {\begin{aligned}\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}{\text{,}}\quad u=\cos {(\theta )}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f198fb3b2b85e8390336ee4c12c74b1106dc6e3c)
![{\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)
![{\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ce8e663abfaa4484542357c4b2320ac8d33408c1)