Integrate the function `e^x(sinx+cosx)`

Answer

To integrate the function \( e^x (\sin x + \cos x) \), we can use the integration formula for functions of the form \( e^x f(x) + e^x f'(x) \). Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Identify the Function**: We need to integrate the function \( e^x (\sin x + \cos x) \).

2. **Recall the Integration Formula**: The integral of the function \( e^x f(x) \) where \( f(x) \) is a differentiable function can be expressed as:
\[ \int e^x f(x) \, dx = e^x f(x) + C \]
where \( C \) is the constant of integration.

3. **Differentiate \( f(x) \)**: In our case, we can take \( f(x) = \sin x + \cos x \). Now, we need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx} (\sin x + \cos x) = \cos x - \sin x \]

4. **Apply the Integration Formula**: Now, we can apply the integration formula:
\[ \int e^x (\sin x + \cos x) \, dx = e^x (\sin x + \cos x) + C \]

5. **Final Answer**: Therefore, the integral of \( e^x (\sin x + \cos x) \) is:
\[ e^x (\sin x + \cos x) + C \]