`inte^(x)(1-cotx+cot^(2)x)dx=`

Options

A. `e^(x)cotx+C`
B. `-e^(x)cotx+C`
C. `e^(x)"cosec "x+C`
D. `-e^(x)"cosec " x+C`

Explanation

To solve the integral \( \int e^x (1 - \cot x + \cot^2 x) \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We start with the expression inside the integral:
\[ 1 - \cot x + \cot^2 x \]
We can recognize that:
\[ \cot^2 x + 1 = \csc^2 x \]
Thus, we can rewrite the integrand:
\[ 1 - \cot x + \cot^2 x = 1 + \cot^2 x - \cot x = \csc^2 x - \cot x \]
So, the integral becomes:
\[ \int e^x (\csc^2 x - \cot x) \, dx \]

### Step 2: Separate the integral
We can separate the integral into two parts:
\[ \int e^x \csc^2 x \, dx - \int e^x \cot x \, dx \]

### Step 3: Use integration by parts
For the integral \( \int e^x \cot x \, dx \), we can use integration by parts. Let:
- \( u = \cot x \) and \( dv = e^x \, dx \)
Then, we differentiate and integrate:
- \( du = -\csc^2 x \, dx \)
- \( v = e^x \)

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \int e^x \cot x \, dx = e^x \cot x - \int e^x (-\csc^2 x) \, dx \]
This simplifies to:
\[ e^x \cot x + \int e^x \csc^2 x \, dx \]

### Step 4: Substitute back into the integral
Now, substituting back into our separated integrals:
\[ \int e^x \csc^2 x \, dx - \left( e^x \cot x + \int e^x \csc^2 x \, dx \right) \]
This leads to:
\[ \int e^x \csc^2 x \, dx - e^x \cot x - \int e^x \csc^2 x \, dx = -e^x \cot x \]

### Step 5: Final answer
Thus, the final result of the integral is:
\[ -e^x \cot x + C \]