`int_(0)^(pi//4) e^(x) (tan x+ sec^(2)x) dx`

Answer

To solve the integral

\[ \int_{0}^{\frac{\pi}{4}} e^{x} \left( \tan x + \sec^{2} x \right) dx, \]

we can use the integration formula for the product of an exponential function and the derivative of another function. The formula states:

\[ \int e^{x} (f(x) + f'(x)) \, dx = e^{x} f(x) + C, \]

where \( f(x) \) is a differentiable function and \( f'(x) \) is its derivative.

### Step-by-step Solution:

1. **Identify \( f(x) \)**:
Here, we can let \( f(x) = \tan x \). The derivative of \( f(x) \) is \( f'(x) = \sec^{2} x \).

2. **Apply the Integration Formula**:
According to the formula, we can rewrite the integral as:
\[ \int e^{x} \left( \tan x + \sec^{2} x \right) dx = e^{x} \tan x + C. \]

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

4. **Calculate the Upper Limit**:
First, we calculate the upper limit:
\[ e^{\frac{\pi}{4}} \tan\left(\frac{\pi}{4}\right) = e^{\frac{\pi}{4}} \cdot 1 = e^{\frac{\pi}{4}}. \]

5. **Calculate the Lower Limit**:
Next, we calculate the lower limit:
\[ e^{0} \tan(0) = 1 \cdot 0 = 0. \]

6. **Subtract the Lower Limit from the Upper Limit**:
Now we subtract the lower limit from the upper limit:
\[ e^{\frac{\pi}{4}} - 0 = e^{\frac{\pi}{4}}. \]

### Final Answer:
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{4}} e^{x} \left( \tan x + \sec^{2} x \right) dx = e^{\frac{\pi}{4}}. \]