5.4 Indefinite Integrals and the Net Change Theorem/33

${\begin{aligned}\int _{1}^{4}{\sqrt {\frac {5}{x}}}dy&=\int _{1}^{4}({\frac {\sqrt {5}}{\sqrt {x}}})dx=5^{\frac {1}{2}}\int _{1}^{4}x^{\frac {-1}{2}}dx\\[2ex]&={\sqrt {5}}\times 2x^{\frac {1}{2}}{\bigg |}_{1}^{4}={\sqrt {5}}\times 2{\sqrt {x}}{\bigg |}_{1}^{4}=2{\sqrt {5x}}{\bigg |}_{1}^{4}\\[2ex]&=2{\sqrt {5\times 4}}-2{\sqrt {5\times 1}}{\bigg |}_{1}^{4}\\[2ex]&=2{\sqrt {20}}-2{\sqrt {5}}{\bigg |}_{1}^{4}=4{\sqrt {5}}-2{\sqrt {5}}{\bigg |}_{1}^{4}=2{\sqrt {5}}\end{aligned}}$