`inte^(x) ( 1- cot x + cot^(2) x) dx` is eqal to

Options

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

Answer

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

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int e^x (1 - \cot x + \cot^2 x) \, dx \]

### Step 2: Recognize a Trigonometric Identity
We know that:
\[ 1 + \cot^2 x = \csc^2 x \]
Thus, we can rewrite the integral as:
\[ I = \int e^x \left( \csc^2 x - \cot x \right) \, dx \]

### Step 3: Use Integration by Parts
We can use the formula for integration involving the derivative of a function:
\[ \int e^x f(x) \, dx = e^x f(x) - \int e^x f'(x) \, dx \]
Here, let \( f(x) = -\cot x \). Then, the derivative \( f'(x) \) is:
\[ f'(x) = -\csc^2 x \]

### Step 4: Apply the Integration Formula
Now, applying the integration formula:
\[ I = e^x (-\cot x) - \int e^x (-\csc^2 x) \, dx \]
This simplifies to:
\[ I = -e^x \cot x + \int e^x \csc^2 x \, dx \]

### Step 5: Solve the Remaining Integral
The integral \( \int e^x \csc^2 x \, dx \) can be recognized as similar to the original integral. Thus, we can denote it as \( J \):
\[ J = \int e^x \csc^2 x \, dx \]
So, we have:
\[ I = -e^x \cot x + J \]

### Step 6: Solve for the Integral
Now we can express \( I \) in terms of itself:
\[ I = -e^x \cot x + I \]
Subtract \( I \) from both sides:
\[ 0 = -e^x \cot x + J - I \]
This gives us:
\[ I = -e^x \cot x + \int e^x \csc^2 x \, dx \]

### Step 7: Final Result
Thus, we can conclude that:
\[ I = -e^x \cot x + C \]
where \( C \) is the constant of integration.

### Final Answer
The final result of the integral is:
\[ \int e^x (1 - \cot x + \cot^2 x) \, dx = -e^x \cot x + C \]