5.5 The Substitution Rule/31: Difference between revisions
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\int \frac{\cos{(x)}}{\sin^2{(x)}}dx | \int \frac{\cos{(x)}}{\sin^2{(x)}}dx | ||
</math> | </math> | ||
<math> | <math> | ||
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u &=\sin{(x)} \\[2ex] | u &=\sin{(x)} \\[2ex] | ||
du &= \cos{(x)}dx | du &= \cos{(x)}dx | ||
\end{align} | |||
</math> | |||
<math> | |||
\begin{align} | |||
\int \frac{\cos{(x)}}{\sin^2{(x)}}dx &= \int\frac{1}{u^2}du = \int u^{-2}du \\[2ex] | |||
&= u^{-1} + C \\[2ex] | |||
&= \frac{-1}{\sin{(x)}} + C | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 15:53, 4 October 2022
![{\displaystyle {\begin{aligned}u&=\sin {(x)}\\[2ex]du&=\cos {(x)}dx\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c9371c8d0e54e3be6eb14d031c53b463708cc897)
![{\displaystyle {\begin{aligned}\int {\frac {\cos {(x)}}{\sin ^{2}{(x)}}}dx&=\int {\frac {1}{u^{2}}}du=\int u^{-2}du\\[2ex]&=u^{-1}+C\\[2ex]&={\frac {-1}{\sin {(x)}}}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/01ea9bf3de6b2dbd1fe80ef10e3d5c61beae7f8d)