5.5 The Substitution Rule/61: Difference between revisions
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= <math>\frac{3}{2}\sqrt[3]{1+2x}\bigg|_{0}^{13}</math> | = <math>\frac{3}{2}\sqrt[3]{1+2x}\bigg|_{0}^{13}</math> | ||
= <math>\frac{3}{2}\sqrt[3]{1+2x* 13}-\frac{3}{2}\sqrt[3]{1+2*0}</math> | = <math>\frac{3}{2}\sqrt[3]{1+2x* 13}-\frac{3}{2}\sqrt[3]{1+2*0}</math> | ||
= <math>\ 3 </math> | = <math>\ 3 </math> \\[2ex] | ||
<math> | |||
\int_{0}^{13}\frac{dx}{\sqrt[3]{(1+2x)^2}}\,dx | |||
</math> | |||
<math> | |||
\begin{align} | |||
u &= 1+2x \\[2ex] | |||
du &= 2dx \\[2ex] | |||
\frac{1}{2}du &= dx \\[2ex] | |||
\end{align} | |||
</math> | |||
New upper limit: <math>\pi = 1+2(13) = 27</math><br> | |||
New lower limit: <math>0 = 1+2(0) = 1</math> | |||
<math> | |||
\begin{align} | |||
\int_{0}^{13}\frac{dx}{\sqrt[3]{(1+2x)^2}}\,dx &= \int_{0}^{\sqrt{\pi}} (xdx)\cos{(x^2)} \\[2ex] | |||
&= \int_{0}^{\pi} \left(\frac{1}{2}du\right)\cos{(u)} = \frac{1}{2}\int_{0}^{\pi} \cos{(u)}du \\[2ex] | |||
&= \frac{1}{2}\sin{(u)}\bigg|_{0}^{\pi} \\[2ex] | |||
&= \frac{1}{2}\sin{(\pi)} - \frac{1}{2}\sin{(0)} \\[2ex] | |||
&= 0 | |||
\end{align} | |||
</math> | |||
Revision as of 03:53, 22 September 2022
= = = = = = = = = \\[2ex]
![{\displaystyle \int _{0}^{13}{\frac {dx}{\sqrt[{3}]{(1+2x)^{2}}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/366746df0224b60295f774b7c9869a3c355329ff)
![{\displaystyle {\frac {1}{2}}\int _{0}^{13}{\frac {1}{\sqrt[{3}]{t^{2}}}}dt}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1d54296bace91b1eb68b7a819d224cf901cf5b7d)
![{\displaystyle {\frac {1}{2}}3{\sqrt[{3}]{t}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/16a3b59a69048f671074fd0201320c2ffb6b137e)
![{\displaystyle {\frac {1}{2}}3{\sqrt[{3}]{1+2x}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/20415fd42cdb3705a79026c6b206e0d8eae227e1)
![{\displaystyle {\frac {3}{2}}{\sqrt[{3}]{1+2x}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/af311e7d8c3cf04c63a34118cdc43ae40922fd8a)
![{\displaystyle {\frac {3}{2}}{\sqrt[{3}]{1+2x}}{\bigg |}_{0}^{13}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a499de07f4b55f145653fc30d4d5ebc73aaa144d)
![{\displaystyle {\frac {3}{2}}{\sqrt[{3}]{1+2x*13}}-{\frac {3}{2}}{\sqrt[{3}]{1+2*0}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6b11b0f2aaf612f19578317fe0894d48b4ce3756)
![{\displaystyle \int _{0}^{13}{\frac {dx}{\sqrt[{3}]{(1+2x)^{2}}}}\,dx}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a0aa7b21c3149c3850f7828b1b0eea1cd73810c2)
![{\displaystyle {\begin{aligned}u&=1+2x\\[2ex]du&=2dx\\[2ex]{\frac {1}{2}}du&=dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a737fc278965117d70e906c0f6291b759f13ac66)
New upper limit:
New lower limit:
![{\displaystyle {\begin{aligned}\int _{0}^{13}{\frac {dx}{\sqrt[{3}]{(1+2x)^{2}}}}\,dx&=\int _{0}^{\sqrt {\pi }}(xdx)\cos {(x^{2})}\\[2ex]&=\int _{0}^{\pi }\left({\frac {1}{2}}du\right)\cos {(u)}={\frac {1}{2}}\int _{0}^{\pi }\cos {(u)}du\\[2ex]&={\frac {1}{2}}\sin {(u)}{\bigg |}_{0}^{\pi }\\[2ex]&={\frac {1}{2}}\sin {(\pi )}-{\frac {1}{2}}\sin {(0)}\\[2ex]&=0\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ba9f4898a7525da41d6642d168c9209769d053a3)