<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] $$\int \frac{ \sqrt{x}}{1 - \sqrt{x}} = - \int \frac{\sqrt{x}}{\sqrt{x}-1} = - \int \frac{ \sqrt{x}-1+1}{\sqrt{x}-1}= - \int dx - \int \frac{dx}{\sqrt{x}-1}$$
But, $ d( \sqrt{x} - 1) = \frac{1}{2 \sqrt{x} } dx $. Hence
$$\int \frac{dx}{\sqrt{x}-1} = 2 \int \frac{\sqrt{x} d( \sqrt{x}-1)}{\sqrt{x}-1}=2 \int \frac{(\sqrt{x}-1+1) d( \sqrt{x}-1)}{\sqrt{x}-1} = 2 \int d( \sqrt{x}-1) + 2 \int \frac{ d(\sqrt{x}-1)}{\sqrt{x}-1} = 2 (\sqrt{x}-1) + 2 \ln(\sqrt{x}-1)$$
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