202202070744 Solution to 1997-CE-AMATH-II-2

Find $\displaystyle{\int x\sqrt{x-1}\,\mathrm{d}x}$ .

[Hint: Let $u=x-1$ .]

Solution.

$\begin{aligned} \textrm{Let } & u = x-1 \\ \textrm{then } & x = u+1 \\ \textrm{and }&\mathrm{d}x = \mathrm{d}u\\ \end{aligned}$

$\begin{aligned} &\quad \int x\sqrt{x-1}\,\mathrm{d}x\\ & = \int (u+1)(u^{\frac{1}{2}})\,\mathrm{d}u \\ & = \int (u^{\frac{3}{2}}+u^{\frac{1}{2}})\,\mathrm{d}u \\ & = \frac{2u^{\frac{5}{2}}}{5} + \frac{2u^{\frac{3}{2}}}{3} + C\textrm{ for some constant }C \\ & = \frac{2(x-1)^{\frac{5}{2}}}{5} + \frac{2(x-1)^{\frac{3}{2}}}{3} + C \\ \end{aligned}$

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