5.4 Indefinite Integrals and the Net Change Theorem/39: Difference between revisions
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&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg|_{1}^{64} | &= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg|_{1}^{64} | ||
2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6}) | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
= <math>2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})</math> | = <math>2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})</math> | ||
Revision as of 16:22, 21 September 2022
![{\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]&=\left[{\frac {x^{\frac {1}{2}}}{\frac {1}{2}}}+{\frac {x^{\frac {5}{6}}}{\frac {5}{6}}}\right]_{1}^{64}=\left[2x^{\frac {1}{2}}+{\frac {6}{5}}x^{\frac {5}{6}}\right]_{1}^{64}\\[2ex]&=2(x)^{\frac {1}{2}}+{\frac {6}{5}}(x)^{\frac {5}{6}}{\bigg |}_{1}^{64}2(64)^{\frac {1}{2}}+{\frac {6}{5}}(64)^{\frac {5}{6}}-(2(1)^{\frac {1}{2}}+{\frac {6}{5}}(1)^{\frac {5}{6}})\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/629dcce7e0561d13594259e7e3c55dec31d67aa4)
= 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}}