`int (2+cot x-cosec^(2) x)e^(x) dx`

Answer

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

### Step 1: Identify the function and its derivative
We notice that the expression \( 2 + \cot x \) has a derivative that includes \( -\csc^2 x \). Specifically, the derivative of \( \cot x \) is \( -\csc^2 x \). Therefore, we can express our integral in terms of a function whose derivative we can recognize.

### Step 2: Rewrite the integral
We can rewrite the integral as:
\[ \int (2 + \cot x - \csc^2 x) e^x \, dx = \int (2 + \cot x) e^x \, dx - \int \csc^2 x \cdot e^x \, dx \]

### Step 3: Recognize the integration formula
We can use the formula for integration by parts or the formula for integrating \( e^x f(x) \) where \( f(x) \) is a function and \( f'(x) \) is its derivative. Here, we can see that:
- Let \( f(x) = 2 + \cot x \)
- Then \( f'(x) = -\csc^2 x \)

### Step 4: Apply the integration formula
Using the integration formula:
\[ \int e^x f(x) \, dx = e^x f(x) + C \]
we can apply it to our integral:
\[ \int e^x (2 + \cot x) \, dx = e^x (2 + \cot x) + C \]

### Step 5: Combine the results
Now, we can combine our results:
\[ \int (2 + \cot x - \csc^2 x) e^x \, dx = e^x (2 + \cot x) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int (2 + \cot x - \csc^2 x) e^x \, dx = e^x (2 + \cot x) + C \]

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