`int cosec^(2)x. sqrt(cot) dx`

Answer

To solve the integral \( \int \csc^2 x \sqrt{\cot x} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \cot x \). Then, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = -\csc^2 x \implies dt = -\csc^2 x \, dx \implies dx = -\frac{dt}{\csc^2 x} \]

### Step 2: Rewrite the Integral
Substituting \( t \) and \( dx \) into the integral, we have:
\[ \int \csc^2 x \sqrt{\cot x} \, dx = \int \csc^2 x \sqrt{t} \left(-\frac{dt}{\csc^2 x}\right) \]
This simplifies to:
\[ -\int \sqrt{t} \, dt \]

### Step 3: Integrate
Now we can integrate \( -\int \sqrt{t} \, dt \):
\[ -\int t^{1/2} \, dt = -\left(\frac{t^{3/2}}{3/2}\right) + C = -\frac{2}{3} t^{3/2} + C \]

### Step 4: Substitute Back
Now, we substitute back \( t = \cot x \):
\[ -\frac{2}{3} (\cot x)^{3/2} + C \]

### Final Answer
Thus, the integral \( \int \csc^2 x \sqrt{\cot x} \, dx \) is:
\[ -\frac{2}{3} \cot^{3/2} x + C \]

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