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

Concept: U Substitution — we rewrite the denominator in a form that cancels the numerator after a clever substitution.

Let $x = t^2$, so $dx = 2t\,dt$ and $\sqrt{x} = t$. Then

Integrate: …

Why U Substitution Works Here

When you see a square root inside a rational function, your first instinct should be to remove the root by substituting the variable inside it. The expression $\sqrt{x}$ is the troublemaker — it makes the denominator $x - \sqrt{x}$ look like a mix of two different powers of $x$. If we set $t = \sqrt{x}$, then $x = t^2$, and suddenly everything becomes a clean rational function in $t$. The derivative $dx = 2t\,dt$ will also give us a factor that cancels nicely.

The deeper reason this works: the integrand is a rational function of $\sqrt{x}$, which is a classic case for the substitution $t = \sqrt{x}$. This transforms the integral into a standard form that we can handle with partial fractions or a simple logarithm.

Step-by-Step Solution

1. Set up the substitution.

   Let $t = \sqrt{x}$. Then $x = t^2$, and differentiating gives $dx = 2t\,dt$.

   The integrand $\frac{1}{x - \sqrt{x}}$ becomes $\frac{1}{t^2 - t}$.

2. Rewrite the integral in terms of $t$.

3. Simplify the integrand. Cancel the common factor $t$ (provided $t \neq 0$, i.e., $x \neq 0$):

So the integral reduces to:

$\int \frac{2}{t-1}\,dt.$ …

Method: Rationalising Substitution $t=\sqrt{x}$

Use this when the integrand is a rational expression built from $x$ and $\sqrt{x}$: replacing the root with a new variable turns the mixed powers into an ordinary rational function.

Steps

Step 1: Let the root be the new variable.

Set $t=\sqrt{x}$, so $x=t^2$ and $dx=2t\,dt$. Every $\sqrt{x}$ becomes $t$ and every $x$ becomes $t^2$, eliminating the awkward half-powers.

Step 2: Simplify the resulting rational function. …

Common Mistakes

Mistake 1: Forgetting the $dx=2t\,dt$ factor.

Why it's wrong: with $x=t^2$, $dx$ is $2t\,dt$, not $dt$; missing it drops a whole factor of $2t$. Correct approach: differentiate $x=t^2$ carefully before substituting.

Mistake 2: Cancelling $\dfrac{2t}{t(t-1)}$ incorrectly. …