`int cot^(3) x. cosec^(2) x dx `

Answer

To solve the integral \( \int \cot^3 x \csc^2 x \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \cot x \). Then, we need to find \( dt \):
\[ \frac{dt}{dx} = -\csc^2 x \implies dt = -\csc^2 x \, dx \]
From this, we can express \( dx \):
\[ dx = -\frac{dt}{\csc^2 x} \]

### Step 2: Rewrite the Integral
Now, we can rewrite the integral in terms of \( t \):
\[ \int \cot^3 x \csc^2 x \, dx = \int t^3 (-dt) = -\int t^3 \, dt \]

### Step 3: Integrate
Now we can integrate:
\[ -\int t^3 \, dt = -\left(\frac{t^4}{4}\right) + C = -\frac{t^4}{4} + C \]

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

### Final Answer
Thus, the final result of the integral is:
\[ -\frac{\cot^4 x}{4} + C \]