5.3 The Fundamental Theorem of Calculus/29

${\begin{aligned}\int _{1}^{9}{\frac {x-1}{\sqrt {x}}}\,dx&=\int _{1}^{9}\left({\frac {x}{\sqrt {x}}}-{\frac {1}{\sqrt {x}}}\right)dx=\int _{1}^{9}\left({\frac {x}{x^{1/2}}}-{\frac {1}{x^{1/2}}}\right)dx=\int _{1}^{9}\left(x^{1/2}-x^{-1/2}\right)dx\\[2ex]&={\frac {2x^{3/2}}{3}}-2x^{1/2}{\bigg |}_{1}^{9}\\[2ex]&=\left[{\frac {2(9)^{3/2}}{3}}-2(9)^{1/2}\right]-\left[{\frac {2(1)^{3/2}}{3}})-2(1)^{1/2}\right]\\[2ex]&=\left[{\frac {54}{3}}-6\right]-\left[{\frac {2}{3}}-2\right]={\frac {52}{3}}-4\\[2ex]&={\frac {40}{3}}\end{aligned}}$