`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 can use integration by parts. Here’s the step-by-step solution:

### Step 1: Identify the parts for integration by parts
We can rewrite the integral as:
\[ \int e^x (4x^2 + 8x + 3) \, dx \]
Let \( u = 4x^2 + 8x + 3 \) and \( dv = e^x \, dx \). Then, we need to find \( du \) and \( v \):
\[ du = (8x + 8) \, dx \]
\[ v = e^x \]

### Step 2: Apply integration by parts
Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we have:
\[ \int e^x (4x^2 + 8x + 3) \, dx = e^x (4x^2 + 8x + 3) - \int e^x (8x + 8) \, dx \]

### Step 3: Simplify the remaining integral
Now we need to evaluate \( \int e^x (8x + 8) \, dx \). We can again use integration by parts:
Let \( u = 8x + 8 \) and \( dv = e^x \, dx \). Then:
\[ du = 8 \, dx \]
\[ v = e^x \]
Applying integration by parts again:
\[ \int e^x (8x + 8) \, dx = e^x (8x + 8) - \int e^x (8) \, dx \]
\[ = e^x (8x + 8) - 8 \int e^x \, dx \]
\[ = e^x (8x + 8) - 8 e^x + C \]

### Step 4: Substitute back into the original integral
Now substituting this back into our original equation:
\[ \int e^x (4x^2 + 8x + 3) \, dx = e^x (4x^2 + 8x + 3) - (e^x (8x + 8) - 8 e^x) \]
\[ = e^x (4x^2 + 8x + 3) - e^x (8x + 8) + 8 e^x \]

### Step 5: Combine like terms
Now we combine like terms:
\[ = e^x (4x^2 + 8x + 3 - 8x - 8 + 8) \]
\[ = e^x (4x^2 + 3) \]

### Final Result
Thus, the final result is:
\[ \int e^x (4x^2 + 8x + 3) \, dx = e^x (4x^2 + 3) + C \]