∫ 1 4 5 x d y = ∫ 1 4 5 x d x = 5 1 2 ∫ 1 4 x − 1 2 d x = 5 × 2 x 1 2 | 1 4 = 5 × 2 x | 1 4 = 2 5 x | 1 4 = 2 5 × 4 − 2 5 × 1 = 2 20 − 2 5 = 4 5 − 2 5 = 2 5 {\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]&={\sqrt {5}}\times 2x^{\frac {1}{2}}{\bigg |}_{1}^{4}={\sqrt {5}}\times 2{\sqrt {x}}{\bigg |}_{1}^{4}=2{\sqrt {5x}}{\bigg |}_{1}^{4}\\[2ex]&=2{\sqrt {5\times 4}}-2{\sqrt {5\times 1}}\\[2ex]&=2{\sqrt {20}}-2{\sqrt {5}}=4{\sqrt {5}}-2{\sqrt {5}}=2{\sqrt {5}}\end{aligned}}}
![{\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]&={\sqrt {5}}\times 2x^{\frac {1}{2}}{\bigg |}_{1}^{4}={\sqrt {5}}\times 2{\sqrt {x}}{\bigg |}_{1}^{4}=2{\sqrt {5x}}{\bigg |}_{1}^{4}\\[2ex]&=2{\sqrt {5\times 4}}-2{\sqrt {5\times 1}}\\[2ex]&=2{\sqrt {20}}-2{\sqrt {5}}=4{\sqrt {5}}-2{\sqrt {5}}=2{\sqrt {5}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ab6be5e5f22e3fe9fc9ad83649073f67393fc028)