Answer:
[tex] \frac{2}{3} \bigg[ x\sqrt{ {x}} + (x - 1)\sqrt{ {(x - 1)}} \bigg] + c[/tex]
Step-by-step explanation:
[tex] \int \frac{1}{ \sqrt{x} - \sqrt{x - 1} } dx \\ \\ = \int \frac{ \sqrt{x} + \sqrt{x - 1}}{ (\sqrt{x} - \sqrt{x - 1} )( \sqrt{x} + \sqrt{x - 1} )} dx \\ \\ = \int \frac{ \sqrt{x} + \sqrt{x - 1}}{ (\sqrt{x})^{2} - (\sqrt{x - 1} )^{2}} dx \\ \\ = \int \frac{ \sqrt{x} + \sqrt{x - 1}}{ x - (x - 1 )} dx \\ \\ = \int \frac{ \sqrt{x} + \sqrt{x - 1}}{ x - x + 1} dx \\ \\ = \int \frac{ \sqrt{x} + \sqrt{x - 1}}{ 1} dx \\ \\ = \int (\sqrt{x} + \sqrt{x - 1}) dx \\ \\ = \int \sqrt{x} \: dx+ \int\sqrt{x - 1} \: dx \\ \\ = \int {x}^{ \frac{1}{2} } \: dx+ \int {(x - 1)}^{ \frac{1}{2} } \: dx \\ \\ = \frac{ {x}^{ \frac{3}{2} } }{ \frac{3}{2} } + \frac{ {(x - 1)}^{ \frac{3}{2} } }{ \frac{3}{2} } + c \\ \\ = \frac{2}{3} {x}^{ \frac{3}{2} } + \frac{2}{3} {(x - 1)}^{ \frac{3}{2} } + c \\ \\ = \frac{2}{3} \bigg[ \sqrt{ {x}^{3} } + \sqrt{ {(x - 1)}^{3} } \bigg] + c \\ \\ = \bold{\purple {\frac{2}{3} \bigg[ x\sqrt{ {x}} + (x - 1)\sqrt{ {(x - 1)}} \bigg] + c}} [/tex]
