5.4 Indefinite Integrals and the Net Change Theorem/39: Difference between revisions
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<math> | <math> | ||
\begin{align} | |||
\int_{1}^{64}\frac{1+\sqrt[3]{x}}\sqrt{x}dx = \int_{1}^{64}\left(\frac{1}{x^{1/2}} + \frac{x^{1/3}}{x^{1/2}}\right)dx | \int_{1}^{64}\frac{1+\sqrt[3]{x}}\sqrt{x}dx &= \int_{1}^{64}\left(\frac{1}{x^{1/2}} + \frac{x^{1/3}}{x^{1/2}}\right)dx | ||
= \int_{1}^{64}\left(x^{-1/2}+x^{\frac{1}{3}-{\frac{1}{2}}}\right)dx = \int_{1}^{64}\left(x^{-\frac{1}{2}}+x^{-\frac{1}{6}}\right)dx | = \int_{1}^{64}\left(x^{-1/2}+x^{\frac{1}{3}-{\frac{1}{2}}}\right)dx = \int_{1}^{64}\left(x^{-\frac{1}{2}}+x^{-\frac{1}{6}}\right)dx \\[2ex] | ||
= \int_{1}^{64}\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+ \frac{x^{\frac{5}{6}}}{\frac{5}{6}} = \int_{1}^{64}2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6} | = \int_{1}^{64}\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+ \frac{x^{\frac{5}{6}}}{\frac{5}{6}} = \int_{1}^{64}2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6} | ||
\end{align} | |||
</math> | </math> | ||
Revision as of 16:19, 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}^{64}{\frac {1+{\sqrt[{3}]{x}}}{\sqrt {x}}}dx&=\int _{1}^{64}\left({\frac {1}{x^{1/2}}}+{\frac {x^{1/3}}{x^{1/2}}}\right)dx=\int _{1}^{64}\left(x^{-1/2}+x^{{\frac {1}{3}}-{\frac {1}{2}}}\right)dx=\int _{1}^{64}\left(x^{-{\frac {1}{2}}}+x^{-{\frac {1}{6}}}\right)dx\\[2ex]=\int _{1}^{64}{\frac {x^{\frac {1}{2}}}{\frac {1}{2}}}+{\frac {x^{\frac {5}{6}}}{\frac {5}{6}}}=\int _{1}^{64}2x^{\frac {1}{2}}+{\frac {6}{5}}x^{\frac {5}{6}}\end{aligned}}}
=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg|_{1}^{64}} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})}
= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 16+38.4 - (2+1.2)}
= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 54.4 - 3.2} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 51.2}
= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{256}{5}}