If $\int \frac{d x}{\sqrt{x}-\sqrt{x-1}}=\lambda\left(x^{\frac{3}{2}}+(x-1)^{\frac{3}{2}}\right)+C$, then the value of $\lambda$ is

$\frac{2}{3}$

$I =\int \frac{1}{\sqrt{x}-\sqrt{x-1}} \times \frac{\sqrt{x}+\sqrt{x-1}}{\sqrt{x}+\sqrt{x-1}} d x$

$I =\int \frac{\sqrt{x}+\sqrt{x-1}}{x-x+1} d x=\int \sqrt{x}+\sqrt{x-1} d x$

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

on comparison $\lambda=\frac{2}{3}$