5.4 Indefinite Integrals and the Net Change Theorem/33: Difference between revisions
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\int_{1}^{4}\sqrt{\frac{5}{x}}dy &= \int_{1}^{4}\frac{\sqrt{5}}{\sqrt{x}}dx = 5^\frac{1}{2}\int_{1}^{4}x^{-\frac{1}{2}}dx\\[2ex] | \int_{1}^{4}\sqrt{\frac{5}{x}}dy &= \int_{1}^{4}\frac{\sqrt{5}}{\sqrt{x}}dx = 5^\frac{1}{2}\int_{1}^{4}x^{-\frac{1}{2}}dx\\[2ex] | ||
&= 2\sqrt{5}x^{\frac{1}{2}}\bigg|_{1}^{4} = 2\sqrt{5}\sqrt{4}-2\sqrt{5}{\sqrt{1}} | &= 2\sqrt{5}x^{\frac{1}{2}}\bigg|_{1}^{4} \\[2ex] | ||
&= 2\sqrt{5}\sqrt{4}-2\sqrt{5}{\sqrt{1}} = 4\sqrt{5}-2\sqrt{5} \\[2ex] | |||
&= 2\sqrt{5} | &= 2\sqrt{5} | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Revision as of 15:29, 21 September 2022
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{1}^{4}{\sqrt {\frac {5}{x}}}dy&=\int _{1}^{4}{\frac {\sqrt {5}}{\sqrt {x}}}dx=5^{\frac {1}{2}}\int _{1}^{4}x^{-{\frac {1}{2}}}dx\\[2ex]&=2{\sqrt {5}}x^{\frac {1}{2}}{\bigg |}_{1}^{4}\\[2ex]&=2{\sqrt {5}}{\sqrt {4}}-2{\sqrt {5}}{\sqrt {1}}=4{\sqrt {5}}-2{\sqrt {5}}\\[2ex]&=2{\sqrt {5}}\end{aligned}}}