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

Answer

To solve the integral \( \int e^x (\cot x - \csc^2 x) \, dx \), we can break it down into two separate integrals:

\[ \int e^x \cot x \, dx - \int e^x \csc^2 x \, dx \]

### Step 1: Solve \( \int e^x \cot x \, dx \)

To solve \( \int e^x \cot x \, dx \), we will use integration by parts. We choose:

- \( u = \cot x \) (first function)
- \( dv = e^x \, dx \) (second function)

Now we need to find \( du \) and \( v \):

- \( du = -\csc^2 x \, dx \)
- \( v = e^x \)

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[ \int e^x \cot x \, dx = e^x \cot x - \int e^x (-\csc^2 x) \, dx \]

This simplifies to:

\[ \int e^x \cot x \, dx = e^x \cot x + \int e^x \csc^2 x \, dx \]

### Step 2: Combine the integrals

Now we can substitute this back into our original integral:

\[ \int e^x (\cot x - \csc^2 x) \, dx = \left( e^x \cot x + \int e^x \csc^2 x \, dx \right) - \int e^x \csc^2 x \, dx \]

The \( \int e^x \csc^2 x \, dx \) terms cancel out:

\[ \int e^x (\cot x - \csc^2 x) \, dx = e^x \cot x + C \]

### Final Answer

Thus, the final answer for the integral is:

\[ \int e^x (\cot x - \csc^2 x) \, dx = e^x \cot x + C \]