5.5 The Substitution Rule/31: Difference between revisions
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\begin{align} | \begin{align} | ||
\int \frac{\cos{(x)}}{\sin^2{(x)}}dx &= \int\frac{1}{u^2}du = \int u^- | \int \frac{\cos{(x)}}{\sin^2{(x)}}dx &= \int\frac{1}{u^2}du = \int u^(-2)du \\[2ex] | ||
&= u^-1 + C \\[2ex] | &= u^(-1) + C \\[2ex] | ||
&= \frac{-1}{\sin{(x)}} + C | &= \frac{-1}{\sin{(x)}} + C | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Revision as of 15:51, 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/0fd39a1215f280612e2cac3b07a161e77bdb3117)