<s>[INST] How to integrate $\frac{\sqrt{x}}{1-\sqrt{x}}$? How to integrate $\frac{\sqrt{x}}{1-\sqrt{x}}$?
I tried by using integration by parts, but always got sucked. Should be very easy...
 [/INST] Let $u=\sqrt{x}$ and $x=u^2$, $\frac{\sqrt{x}}{1-\sqrt{x}}=-1+\frac{1}{1-\sqrt{x}}=-1+\frac{1}{1-u}$, then
\begin{align}
\int{\frac{\sqrt{x}}{1-\sqrt{x}}dx}&=\int{\left(-1+\frac{1}{1-u}\right)(2udu)}\\
&=-\int{2udu}+2\int{\frac{u}{1-u}du} \\
&=-u^2+2\int{\left(-1+\frac{1}{1-u}\right)du} \\
&=-u^2-2u+2\ln|1-u|+C \\
&=-x-2\sqrt{x}+2\ln|1-\sqrt{x}|+C
\end{align}
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