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**Hint:**according to the question we have to find the integration of $\int {\dfrac{{dx}}{{x - \sqrt x }}} $.

So, first of all we have to let$\sqrt x $= u then we have to differentiate both terms with respect to $x$with the help of the formula that is mentioned below.

**Formula used:**

$\dfrac{d}{{dx}}\sqrt x = \dfrac{1}{{2\sqrt x }}...........................(A)$

Now, we have to put all the value of $\sqrt x $and $dx$in the given expression$\int {\dfrac{{dx}}{{x - \sqrt x }}} $

After that we have to use the integration formula of $\dfrac{1}{{(a + 1)}}da$that is mentioned below.

$\int {\dfrac{1}{{a + 1}}} da$$ = \log \left| {a + 1} \right| + C............................(B)$

**Complete answer:**

Step 1: First of all we have to let the $\sqrt x $=u

Now, differentiate both terms with respect to$x$

$ \Rightarrow \dfrac{d}{{dx}}\sqrt x = \dfrac{d}{{dx}}\left( u \right)$

Now, we have to apply the formula (A) that is mentioned in the solution hint.

$

\Rightarrow \dfrac{1}{{2\sqrt x }} = \dfrac{{du}}{{dx}} \\

\Rightarrow dx = 2\sqrt x du........................(1) \\

$

Step 2: Now, we put the value of $\sqrt x $=u in the expression (1) obtained in step 1

$ \Rightarrow dx = 2udu$

Step 3: Now, we put all values obtained in step 1 and step 2 in the given expression $\int {\dfrac{{dx}}{{x - \sqrt x }}} $

$

\Rightarrow \int {\dfrac{{2udu}}{{(u + {u^2})}}} \\

\Rightarrow \int {\dfrac{{2udu}}{{u(1 + u)}}} \\

\Rightarrow \int {\dfrac{{2du}}{{(1 + u)}}} \\

$

Step 4: Now, we have to apply the formula (B) in the expression mentioned in the step 3.

$ \Rightarrow 2\log \left| {u + 1} \right| + C$

Now, put the value of $u$ in terms of $x$.

$ \Rightarrow 2\log \left| {\sqrt x + 1} \right| + C$

**The integrated values of the given expression $\int {\dfrac{{dx}}{{x - \sqrt x }}} $ $ = 2\log \left| {\sqrt x + 1} \right| + C$.**

**Note:**

It is necessary that we have to let the term $\sqrt x $= u and then we have to find the differentiation of the term we let to simplify the expression.

It is necessary that we have let x as ${u^2}$ and then we have to substitute the value in the given expression to find the integration.

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