`inte^(x+3) dx`

Answer

To solve the integral \( \int e^{(x+3)} \, dx \), we can follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int e^{(x+3)} \, dx \]

### Step 2: Use substitution
Let’s perform a substitution to simplify the integral. We can let:
\[ t = x + 3 \]
Then, the differential \( dx \) becomes:
\[ dx = dt \]

### Step 3: Rewrite the integral in terms of \( t \)
Substituting \( t \) into the integral gives us:
\[ I = \int e^t \, dt \]

### Step 4: Integrate
The integral of \( e^t \) is:
\[ \int e^t \, dt = e^t + C \]
where \( C \) is the constant of integration.

### Step 5: Substitute back to the original variable
Now, we substitute back \( t = x + 3 \):
\[ I = e^{(x + 3)} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int e^{(x+3)} \, dx = e^{(x + 3)} + C \]

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