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> | ||
| Line 7: | Line 7: | ||
\begin{align} | \begin{align} | ||
u &= \sqrt{ | u &= \sqrt{t} \\[2ex] | ||
du &= (\frac{1}{2}\ \frac{1}{\sqrt{t}})\;dt \\[2ex] | |||
du &= \frac{1}{2}\ \frac{1}{\sqrt{t}} | 2du &= \frac{1}{\sqrt{t}}\;dt | ||
2du &= | \end{align} | ||
</math> | |||
<math> | |||
\begin{align} | |||
\int \frac{1}{\sqrt{t}}\cos{(\sqrt{t})} dt &= 2\int \cos {u}\;du \\[2ex] | |||
&= 2 \sin{u}+c \\[2ex] | |||
&= 2 \sin(\sqrt{t}) + c \\[2ex] | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 23:27, 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/0ac506fb861c0050d06ca2164528e03909ec6b6e)
![{\displaystyle {\begin{aligned}\int {\frac {1}{\sqrt {t}}}\cos {({\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/cab9134887b680d923404d9383d7bab4ad18629e)