$\int \frac{u\+1}{\sqrt{u}} \mathit{\ⅆ}u$

$\= \int \left(u^{1\/2}\+u^{-1\/2}\right) \ⅆu$

$\=\frac{u^{3\/2}}{3\/2}\+\frac{u^{1\/2}}{1\/2}$

$\=\frac{2}{3}{\left(x-1\right)}^{3\/2}\+2\sqrt{x-1}$

$\=2\sqrt{x-1}\left(x\+2\right)\/3$

Evaluation

$\int \frac{x}{\sqrt{x-1}} \mathit{\ⅆ}x$ = $\frac{2}{3}\sqrt{x-1}\left(x\+2\right)$$$

$\begin{array}{c}\int \frac{x}{\sqrt{x-1}}\mathit{\ⅆ}x\hfill \\ & \text{=}& 2\int u^2\mathit{\ⅆ}u+2u\hfill & \left[\mathrm{change}\,x-1=u^2\,u\right]\\ & \text{=}& \frac{2}{3}{u}^3+2u\hfill & \left[\mathrm{power}\right]\\ & \text{=}& \frac{2{\left(x-1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}+2\sqrt{x-1}\hfill & \left[\mathrm{revert}\right]\end{array}$

Table 6.6.1(a)   Stepwise solution via Integration Methods tutor

$\mathrm{with}\left(\mathrm{IntegrationTools}\right)\:$

$q≔\int \frac{x}{\sqrt{x-1}} \mathit{\ⅆ}x\:$

$\mathrm{TX}≔u\=\sqrt{x-1}\:$

$Q≔\mathrm{Change}\left(q\,\mathrm{TX}\,u\right)$

$\∫\left(2u^2\+2\right)\ⅆu$

$A≔\mathrm{value}\left(Q\right)$

$\frac{2}{3}{u}^3\+2u$

$\genfrac{}{}{0ex}{}{A}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\mathrm{TX}}$ = $\frac{2}{3}{\left(x-1\right)}^{3\/2}\+2\sqrt{x-1}$$$

Table 6.6.1(b)   Rationalizing substitution via the Change command in the IntegrationTools package