`inte^(x)(4x^(2)+8x+3)dx` equals

Options

A. `(2x+1)^(2)e^(x)+k`
B. `(x+1)^(2)e^(x)+k`
C. `(4x+2)e^(x)+k`
D. none of these

Answer

To solve the integral \( \int e^x (4x^2 + 8x + 3) \, dx \), we will use the property of integration involving the exponential function and its derivative.

### Step-by-Step Solution:

1. **Identify the Function and Its Derivative**:
We have \( f(x) = 4x^2 + 8x + 3 \). To apply the property, we need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(4x^2 + 8x + 3) = 8x + 8 \]

2. **Rearranging the Integral**:
We can express the integral in the form \( \int e^x f(x) \, dx \). Notice that \( f'(x) \) is related to \( 4x^2 + 8x + 3 \) since:
\[ f(x) = 4x^2 + 8x + 3 = \frac{1}{4}(f'(x) + 8) \]

3. **Using the Integration Property**:
According to the property of integration:
\[ \int e^x f(x) \, dx = e^x f(x) + C \]
Here, we can apply this property directly since \( f'(x) \) is present in our function.

4. **Substituting Back**:
Now substituting \( f(x) \) back into the equation gives us:
\[ \int e^x (4x^2 + 8x + 3) \, dx = e^x (4x^2 + 8x + 3) + C \]

5. **Final Answer**:
Thus, the integral evaluates to:
\[ \int e^x (4x^2 + 8x + 3) \, dx = e^x (4x^2 + 8x + 3) + C \]