5.4 Indefinite Integrals and the Net Change Theorem/11: Difference between revisions
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\int\frac{x^3-2\sqrt{x}}{x}dx &= \int\left(\frac{x^3}{x}-\frac{2\sqrt{x}}{x}\right)dx = \int\left(x^2-2x^{\frac{1}{2}-1}\right)dx = \int\left(x^2-2x^{-\frac{1}{2}}\right)dx \\[2ex] | \int\frac{x^3-2\sqrt{x}}{x}dx &= \int\left(\frac{x^3}{x}-\frac{2\sqrt{x}}{x}\right)dx = \int\left(x^2-2x^{\frac{1}{2}-1}\right)dx = \int\left(x^2-2x^{-\frac{1}{2}}\right)dx \\[2ex] | ||
&= \frac{x^3}{3}-\frac{2x^\frac{1}{2}}{\frac{1}{2}}+C \\[2ex] | &= \frac{x^3}{3}-\frac{2x^\frac{1}{2}}{\frac{1}{2}}+C \\[2ex] | ||
&= \frac{ | &= \frac{x^3}{3}-4\sqrt{x}+C | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 19:39, 21 September 2022
![{\displaystyle {\begin{aligned}\int {\frac {x^{3}-2{\sqrt {x}}}{x}}dx&=\int \left({\frac {x^{3}}{x}}-{\frac {2{\sqrt {x}}}{x}}\right)dx=\int \left(x^{2}-2x^{{\frac {1}{2}}-1}\right)dx=\int \left(x^{2}-2x^{-{\frac {1}{2}}}\right)dx\\[2ex]&={\frac {x^{3}}{3}}-{\frac {2x^{\frac {1}{2}}}{\frac {1}{2}}}+C\\[2ex]&={\frac {x^{3}}{3}}-4{\sqrt {x}}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d9036c97e28b6a10bf32c94f735a344417552a31)