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| &= \frac{1}{3}\int \frac{1}{\sqrt{u}}(du) = \frac{1}{3}\int u^{-1/2} du \\[2ex] | | &= \frac{1}{3}\int \frac{1}{\sqrt{u}}(du) = \frac{1}{3}\int u^{-1/2} du \\[2ex] |
| &= \frac{1}{3}\frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C \\[2ex] | | &= \frac{1}{3}\frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C \\[2ex] |
| &= \frac{2}{3}(3ax+bx^3)^{1/2} + C | | &= \frac{2}{3}(3ax+bx^3)^{1/2} + C \\[2ex] |
| &= \frac{2}{3}{\sqrt{3ax+bx^3}} + C | | &= \frac{2}{3}{\sqrt{3ax+bx^3}} + C |
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| \end{align} | | \end{align} |
| </math> | | </math> |
Revision as of 16:08, 14 September 2022
![{\displaystyle {\begin{aligned}u&=3ax+bx^{3}\\[2ex]du&=(3a+3bx^{2})dx\\[2ex]{\frac {1}{3}}du&=(a+bx^{2})dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/02464f0e436c19b198f5cf8d28fa89ba4633df4c)
![{\displaystyle {\begin{aligned}\int {\frac {a+bx^{2}}{\sqrt {3ax+bx^{3}}}}dx&=\int {\frac {1}{\sqrt {3ax+bx^{3}}}}(a+bx^{2})\;dx=\int {\frac {1}{\sqrt {3ax+bx^{3}}}}(a+bx^{2}\;dx)\ \\[2ex]&={\frac {1}{3}}\int {\frac {1}{\sqrt {u}}}(du)={\frac {1}{3}}\int u^{-1/2}du\\[2ex]&={\frac {1}{3}}{\frac {u^{\frac {1}{2}}}{\frac {1}{2}}}+C\\[2ex]&={\frac {2}{3}}(3ax+bx^{3})^{1/2}+C\\[2ex]&={\frac {2}{3}}{\sqrt {3ax+bx^{3}}}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/4e66b85158f8454bb2bcdebe81912ca4ef4a99a9)