Find: `int cosec x (cosecx-cot x) dx`

Answer

To solve the integral \( \int \csc x (\csc x - \cot x) \, dx \), we can break it down step by step.

### Step 1: Rewrite the Integral
We start by distributing \( \csc x \) in the integral:
\[ \int \csc x (\csc x - \cot x) \, dx = \int (\csc^2 x - \csc x \cot x) \, dx \]

### Step 2: Split the Integral
Now, we can split the integral into two separate integrals:
\[ \int \csc^2 x \, dx - \int \csc x \cot x \, dx \]

### Step 3: Integrate Each Part
Now, we will integrate each part using standard integral formulas:

1. The integral of \( \csc^2 x \) is:
\[ \int \csc^2 x \, dx = -\cot x \]

2. The integral of \( \csc x \cot x \) is:
\[ \int \csc x \cot x \, dx = -\csc x \]

### Step 4: Combine the Results
Now, substituting these results back into our expression, we have:
\[ -\cot x - (-\csc x) = -\cot x + \csc x \]

### Step 5: Add the Constant of Integration
Finally, we add the constant of integration \( C \):
\[ -\cot x + \csc x + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \csc x (\csc x - \cot x) \, dx = -\cot x + \csc x + C \]

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