∫ 1 64 1 + x 3 x d x = ∫ 1 64 ( 1 x 1 / 2 + x 1 / 3 x 1 / 2 ) d x = ∫ 1 64 ( x − 1 / 2 + x 1 3 − 1 2 ) d x = ∫ 1 64 ( x − 1 2 + x − 1 6 ) d x = [ x 1 2 1 2 + x 5 6 5 6 ] 1 64 = [ 2 x 1 2 + 6 5 x 5 6 ] 1 64 = 2 ( x ) 1 2 + 6 5 ( x ) 5 6 | 1 64 {\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}\end{aligned}}}
= 2 ( x ) 1 2 + 6 5 ( x ) 5 6 | 1 64 {\displaystyle 2(x)^{\frac {1}{2}}+{\frac {6}{5}}(x)^{\frac {5}{6}}{\bigg |}_{1}^{64}} = 2 ( 64 ) 1 2 + 6 5 ( 64 ) 5 6 − ( 2 ( 1 ) 1 2 + 6 5 ( 1 ) 5 6 ) {\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}})}
= 16 + 38.4 − ( 2 + 1.2 ) {\displaystyle 16+38.4-(2+1.2)}
= 54.4 − 3.2 {\displaystyle 54.4-3.2} = 51.2 {\displaystyle 51.2}
= 256 5 {\displaystyle {\frac {256}{5}}}
![{\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}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/52e3d25730034b0d493f63b6d5617ae0fe7ff227)
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