Chapter 5 - Section 5.5 - Indefinite Integrals and the Substitution Method - Exercises - Page 330: 39

$$\int\frac{1}{x^2}\sqrt{2-\frac{1}{x}}dx=\frac{2}{3}\Big(2-\frac{1}{x}\Big)^{3/2}+C$$

$$A=\int\frac{1}{x^2}\sqrt{2-\frac{1}{x}}dx$$ We set $u=2-\frac{1}{x}$ Then $$du=0-\Big(-\frac{1}{x^2}\Big)dx=\frac{1}{x^2}dx$$ Therefore, $$A=\int\sqrt udu=\int u^{1/2}du$$ $$A=\frac{u^{3/2}}{\frac{3}{2}}+C=\frac{2u^{3/2}}{3}+C$$ $$A=\frac{2}{3}\Big(2-\frac{1}{x}\Big)^{3/2}+C$$