`int_(0)^(pi)e^(x)(tan x + sec^(2)x) dx` = ______.

Options

A. 0
B. 1
C. `-1`
D. `-e^(pi)`

Answer

To solve the integral \( \int_{0}^{\pi} e^{x} (\tan x + \sec^{2} x) \, dx \), we will use the integration technique based on the formula for integrating the product of an exponential function and the sum of a function and its derivative.

### Step-by-step Solution:

1. **Identify the Components**:
We have \( e^{x} \) multiplied by \( \tan x + \sec^{2} x \). Notice that \( \sec^{2} x \) is the derivative of \( \tan x \). Thus, we can let:
- \( f(x) = \tan x \)
- \( f'(x) = \sec^{2} x \)

2. **Recognize the Integral Form**:
The integral can be recognized as fitting the form:
\[ \int e^{x} (f(x) + f'(x)) \, dx \]
which integrates to:
\[ e^{x} f(x) + C \]

3. **Apply the Formula**:
Therefore, we can write:
\[ \int e^{x} (\tan x + \sec^{2} x) \, dx = e^{x} \tan x + C \]

4. **Evaluate the Definite Integral**:
Now we need to evaluate this from \( 0 \) to \( \pi \):
\[ \left[ e^{x} \tan x \right]_{0}^{\pi} \]

5. **Calculate the Upper Limit**:
- At \( x = \pi \):
\[ e^{\pi} \tan(\pi) = e^{\pi} \cdot 0 = 0 \]

6. **Calculate the Lower Limit**:
- At \( x = 0 \):
\[ e^{0} \tan(0) = 1 \cdot 0 = 0 \]

7. **Combine the Results**:
Now substituting the limits into the evaluated integral:
\[ 0 - 0 = 0 \]

### Final Answer:
Thus, the value of the integral \( \int_{0}^{\pi} e^{x} (\tan x + \sec^{2} x) \, dx \) is \( \boxed{0} \).