5.5 The Substitution Rule/21: Difference between revisions
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<math> | <math> | ||
\int \frac{\cos{(\sqrt{t})}}{\sqrt{t}} dt | \int \frac{\cos{(\sqrt{t})}}{\sqrt{t}}\;dt | ||
</math> | </math> | ||
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u &= \sqrt{t} \\[2ex] | u &= \sqrt{t} \\[2ex] | ||
du &= \frac{1}{2}\ \frac{1}{\sqrt{t}} dt \\[2ex] | du &= \frac{1}{2}\ \frac{1}{\sqrt{t}}\;dt \\[2ex] | ||
2du &= \frac{1}{\sqrt{t}} dt | 2du &= \frac{1}{\sqrt{t}}\;dt | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
| Line 17: | Line 17: | ||
\begin{align} | \begin{align} | ||
\int \frac{\cos{(\sqrt{t})}}{\sqrt{t}} dt &= 2\int \cos {u} du \\[2ex] | \int \frac{\cos{(\sqrt{t})}}{\sqrt{t}} dt &= 2\int \cos {u}\;du \\[2ex] | ||
&= 2 \sin{u}+c \\[2ex] | &= 2 \sin{u}+c \\[2ex] | ||
&= 2 \sin(\sqrt{t}) + c \\[2ex] | &= 2 \sin(\sqrt{t}) + c \\[2ex] | ||
Revision as of 23:17, 13 September 2022
![{\displaystyle {\begin{aligned}u&={\sqrt {t}}\\[2ex]du&={\frac {1}{2}}\ {\frac {1}{\sqrt {t}}}\;dt\\[2ex]2du&={\frac {1}{\sqrt {t}}}\;dt\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f2c7aaf645b9ff0f94bca91546ec714be11f250a)
![{\displaystyle {\begin{aligned}\int {\frac {\cos {({\sqrt {t}})}}{\sqrt {t}}}dt&=2\int \cos {u}\;du\\[2ex]&=2\sin {u}+c\\[2ex]&=2\sin({\sqrt {t}})+c\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5110ae213621bbd4566e4e4ac69f9cf3338d53c9)