Evaluate: `int e^(x) (1 + tan x + tan^(2)x)dx`

Answer

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

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

### Step 2: Recognize the structure
Notice that \( 1 + \tan x + \tan^2 x \) can be rewritten using the identity \( 1 + \tan^2 x = \sec^2 x \):
\[ 1 + \tan x + \tan^2 x = \tan x + \sec^2 x \]

### Step 3: Use the formula for integration
We can use the formula for integration:
\[ \int e^x f(x) + e^x f'(x) \, dx = e^x f(x) + C \]
where \( f(x) = \tan x \) and \( f'(x) = \sec^2 x \).

### Step 4: Apply the formula
Now we can apply the formula:
\[ I = \int e^x \left( \tan x + \sec^2 x \right) \, dx = e^x \tan x + C \]

### Step 5: Final result
Thus, the evaluated integral is:
\[ \int e^x (1 + \tan x + \tan^2 x) \, dx = e^x \tan x + C \]