`int cosecx(cosec x + cot x) dx = ?`

Options

A. `cot x - cosec x + C`
B. `-cot x + cosec x + C`
C. `cotx + cosec x + C`
D. `-cot x - cosec + C`

Answer

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

### Step 1: Expand the Integral
We start by distributing \( \csc x \) inside 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 Term
1. **Integrate \( \csc^2 x \)**:
The integral of \( \csc^2 x \) is:
\[ \int \csc^2 x \, dx = -\cot x \]

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

### Step 4: Combine the Results
Now, we combine the results of the two integrals:
\[ -\cot x - \csc x + C \]
where \( C \) is the constant of integration.

### 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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