`int e^x (tan x+1) sec x dx`

Answer

To solve the integral \( \int e^x (\tan x + 1) \sec x \, dx \), we can use the method of integration by parts or recognize it as a form that can be simplified using a known formula.

### Step-by-Step Solution:

1. **Identify the components**:
We can rewrite the integral as:
\[ \int e^x \sec x (\tan x + 1) \, dx \]
This can be separated into two parts:
\[ \int e^x \sec x \tan x \, dx + \int e^x \sec x \, dx \]

2. **Use the integration formula**:
We know that:
\[ \int e^x f(x) \, dx = e^x f(x) - \int e^x f'(x) \, dx \]
Here, we can let \( f(x) = \sec x \).

3. **Calculate \( f'(x) \)**:
The derivative of \( f(x) = \sec x \) is:
\[ f'(x) = \sec x \tan x \]

4. **Apply the integration formula**:
Using the formula, we can compute:
\[ \int e^x \sec x \, dx = e^x \sec x - \int e^x \sec x \tan x \, dx \]

5. **Combine the integrals**:
Now, substituting back into our original integral:
\[ \int e^x \sec x (\tan x + 1) \, dx = \int e^x \sec x \tan x \, dx + \int e^x \sec x \, dx \]
We can substitute the expression we found for \( \int e^x \sec x \, dx \):
\[ = \int e^x \sec x \tan x \, dx + \left( e^x \sec x - \int e^x \sec x \tan x \, dx \right) \]

6. **Simplify**:
Let \( I = \int e^x \sec x \tan x \, dx \):
\[ I + (e^x \sec x - I) = e^x \sec x \]
Thus, we have:
\[ 2I = e^x \sec x \implies I = \frac{1}{2} e^x \sec x \]

7. **Final result**:
Therefore, the integral evaluates to:
\[ \int e^x (\tan x + 1) \sec x \, dx = \frac{1}{2} e^x \sec x + C \]

### Final Answer:
\[ \int e^x (\tan x + 1) \sec x \, dx = \frac{1}{2} e^x \sec x + C \]