`inte^(x)(1+tanx+tan^(2)x)dx=`

Options

A. `e^(x).sec^(2)x+c`
B. `e^(x).secx+c`
C. `e^(x).tan^(2)x+c`
D. `e^(x).tanx+c`

Answer

To solve the integral \( \int e^x (1 + \tan x + \tan^2 x) \, dx \), we can break it down into manageable parts.

### Step-by-Step Solution:

1. **Rewrite the Integral**:
We start by rewriting the integral:
\[ \int e^x (1 + \tan x + \tan^2 x) \, dx = \int e^x \, dx + \int e^x \tan x \, dx + \int e^x \tan^2 x \, dx \]

2. **Integrate the First Term**:
The first term is straightforward:
\[ \int e^x \, dx = e^x + C_1 \]

3. **Integrate the Second Term**:
For the second term \( \int e^x \tan x \, dx \), we can use integration by parts. Let:
- \( u = \tan x \) ⇒ \( du = \sec^2 x \, dx \)
- \( dv = e^x \, dx \) ⇒ \( v = e^x \)

Applying integration by parts:
\[ \int e^x \tan x \, dx = e^x \tan x - \int e^x \sec^2 x \, dx \]

4. **Integrate the Third Term**:
For the third term \( \int e^x \tan^2 x \, dx \), we can express \( \tan^2 x \) in terms of \( \sec^2 x \):
\[ \tan^2 x = \sec^2 x - 1 \]
Thus,
\[ \int e^x \tan^2 x \, dx = \int e^x (\sec^2 x - 1) \, dx = \int e^x \sec^2 x \, dx - \int e^x \, dx \]

5. **Combine the Results**:
Now we combine all the parts:
\[ \int e^x (1 + \tan x + \tan^2 x) \, dx = e^x + \left( e^x \tan x - \int e^x \sec^2 x \, dx \right) + \left( \int e^x \sec^2 x \, dx - e^x \right) \]

Notice that the \( -e^x \) from the third term cancels with the \( +e^x \) from the first term:
\[ = e^x \tan x + C \]

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