Evaluate `int(1-cosx)"cosec"^(2)xdx`

Explanation

To evaluate the integral \( \int (1 - \cos x) \csc^2 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int (1 - \cos x) \csc^2 x \, dx \]
We can separate this into two integrals:
\[ \int \csc^2 x \, dx - \int \cos x \csc^2 x \, dx \]

### Step 2: Evaluate the First Integral
The first integral is:
\[ \int \csc^2 x \, dx \]
The integral of \( \csc^2 x \) is a standard result:
\[ \int \csc^2 x \, dx = -\cot x + C_1 \]

### Step 3: Evaluate the Second Integral
Now we need to evaluate the second integral:
\[ \int \cos x \csc^2 x \, dx \]
Recall that \( \csc^2 x = \frac{1}{\sin^2 x} \), thus:
\[ \int \cos x \csc^2 x \, dx = \int \frac{\cos x}{\sin^2 x} \, dx \]
This can be rewritten as:
\[ \int \cot x \, dx \]
The integral of \( \cot x \) is:
\[ \int \cot x \, dx = \ln |\sin x| + C_2 \]

### Step 4: Combine the Results
Now we can combine the results from the two integrals:
\[ \int (1 - \cos x) \csc^2 x \, dx = -\cot x - \ln |\sin x| + C \]
where \( C = C_1 - C_2 \) is the constant of integration.

### Final Answer
Thus, the final answer is:
\[ -\cot x - \ln |\sin x| + C \]

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