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| \begin{align} | | \begin{align} |
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| u &= 3ax+bx^3 \\[2ex] | | u &= \sin(x) \\[2ex] |
| du &= (3a+3bx^2)dx \\[2ex]
| | du &= \cos(x)\;dx \\[2ex] |
| \frac{1}{3}du &= (a+bx^2)dx \\[2ex]
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| \end{align} | | \end{align} |
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| \begin{align} | | \begin{align} |
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| \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] | | \int \frac{\cos(x)}{\sin(x)}dx &= \int \frac{1}{\sin(x)}\cos(x)\;dx = \int \frac{1}{\sin(x)}(\cos(x)\;dx) \\[2ex] |
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| &= \frac{1}{3}\int \frac{1}{\sqrt{u}}(du) = \frac{1}{3}\int u^{-1/2} du \\[2ex] | | &= \int \frac{1}{{u}}(du) \\[2ex] |
| &= \frac{1}{3}\frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C \\[2ex] | | &= \left| \ln(u) \right| + C \\[2ex] |
| &= \frac{2}{3}(3ax+bx^3)^{1/2} + C \\[2ex] | | &= \left| \ln(\sin(x)) \right| + C \\[2ex] |
| &= \frac{2}{3}{\sqrt{3ax+bx^3}} + C
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| \end{align} | | \end{align} |
| </math> | | </math> |
Latest revision as of 19:18, 20 September 2022
![{\displaystyle {\begin{aligned}u&=\sin(x)\\[2ex]du&=\cos(x)\;dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d86146c5ab9052b4f1c419e803482f36424c663c)
![{\displaystyle {\begin{aligned}\int {\frac {\cos(x)}{\sin(x)}}dx&=\int {\frac {1}{\sin(x)}}\cos(x)\;dx=\int {\frac {1}{\sin(x)}}(\cos(x)\;dx)\\[2ex]&=\int {\frac {1}{u}}(du)\\[2ex]&=\left|\ln(u)\right|+C\\[2ex]&=\left|\ln(\sin(x))\right|+C\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/4d7d1b37435a35563651d48b4689b83e331212f5)