5.5 The Substitution Rule/61: Difference between revisions
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\int_{0}^{13}\frac{1}{\sqrt[3]{(1+2x)^2}}\,dx &= \int_{0}^{13}\frac{1}{\sqrt[3]{(1+2x)^2}}\,(dx) \\[2ex] | \int_{0}^{13}\frac{1}{\sqrt[3]{(1+2x)^2}}\,dx &= \int_{0}^{13}\frac{1}{\sqrt[3]{(1+2x)^2}}\,(dx) \\[2ex] | ||
&= \int_{1}^{27}\frac{1}{\sqrt[3]{ | &= \int_{1}^{27}\frac{1}{\sqrt[3]{u^2}}\left(\frac{1}{2}du\right) = \frac{1}{2}\int_{1}^{27} {u}^{-2/3}du \\[2ex] | ||
&= \frac{1}{2}\sin{(u)}\bigg|_{0}^{\pi} \\[2ex] | &= \frac{1}{2}\sin{(u)}\bigg|_{0}^{\pi} \\[2ex] | ||
&= \frac{1}{2}\sin{(\pi)} - \frac{1}{2}\sin{(0)} \\[2ex] | &= \frac{1}{2}\sin{(\pi)} - \frac{1}{2}\sin{(0)} \\[2ex] | ||
Revision as of 04:04, 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 {1}{\sqrt[{3}]{(1+2x)^{2}}}}\,dx}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e9f38b04c5de29c05e4c1854ebafe81175b77978)
![{\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 {1}{\sqrt[{3}]{(1+2x)^{2}}}}\,dx&=\int _{0}^{13}{\frac {1}{\sqrt[{3}]{(1+2x)^{2}}}}\,(dx)\\[2ex]&=\int _{1}^{27}{\frac {1}{\sqrt[{3}]{u^{2}}}}\left({\frac {1}{2}}du\right)={\frac {1}{2}}\int _{1}^{27}{u}^{-2/3}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/8797881bf4e4465a38f936d3b23a693d77fcb9be)