`int_(0)^(pi//2) e^(x) (sin x + cos x) dx`

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{2}} e^{x} (\sin x + \cos x) \, dx \), we can use integration by parts or a known formula for integrals involving exponentials and trigonometric functions.

### Step-by-Step Solution:

1. **Recognize the Integral Form**:
We have the integral in the form \( \int e^{x} (f(x) + f'(x)) \, dx \) where \( f(x) = \sin x \) and \( f'(x) = \cos x \).

2. **Use the Integration Formula**:
The formula states that:
\[ \int e^{x} (f(x) + f'(x)) \, dx = e^{x} f(x) + C \]
Therefore, we can rewrite our integral:
\[ \int e^{x} (\sin x + \cos x) \, dx = e^{x} \sin x + C \]

3. **Evaluate the Integral with Limits**:
We need to evaluate:
\[ \left[ e^{x} \sin x \right]_{0}^{\frac{\pi}{2}} \]
This means we will substitute the upper limit and lower limit into the expression.

4. **Substituting the Upper Limit**:
First, substitute \( x = \frac{\pi}{2} \):
\[ e^{\frac{\pi}{2}} \sin\left(\frac{\pi}{2}\right) = e^{\frac{\pi}{2}} \cdot 1 = e^{\frac{\pi}{2}} \]

5. **Substituting the Lower Limit**:
Next, substitute \( x = 0 \):
\[ e^{0} \sin(0) = 1 \cdot 0 = 0 \]

6. **Final Calculation**:
Now, subtract the lower limit from the upper limit:
\[ e^{\frac{\pi}{2}} - 0 = e^{\frac{\pi}{2}} \]

Thus, the value of the integral \( \int_{0}^{\frac{\pi}{2}} e^{x} (\sin x + \cos x) \, dx \) is:
\[ \boxed{e^{\frac{\pi}{2}}} \]