Integrate the function: 1 x - x - Vidyadip

Question

Integrate the function: $\frac{1}{{x - \sqrt x }}$

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Answer

Let $I = \int\frac{1}{{x - \sqrt x }}dx$...(i)

Putting $\sqrt x = t$

$ \Rightarrow x = {t^2}$

$\Rightarrow \frac{{dx}}{{dt}} = 2t$

$ \Rightarrow dx = 2tdt$

$\therefore $ From eq. (i),

$I = \int {\frac{1}{{{t^2} - t}}2t} dt$

$= 2\int {\frac{t}{{t\left( {t - 1} \right)}}} dt$

$ = 2\int {\frac{1}{{\left( {t - 1} \right)}}} dt$

= 2 log |t - 1| + c

$ = 2\log \left| {\sqrt x - 1} \right| + c$

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