If `intxe^(2x)dx` is equal to `e^(2x)f(x)+c`, where c is constant of integration, then f(x) is

Options

A. `(3x-1)//4`
B. `(2x+1)//2`
C. `(2x-1)//4`
D. `(x-4)//6`

Answer

To solve the integral \( \int x e^{2x} \, dx \) and express it in the form \( e^{2x} f(x) + c \), we will use the method of integration by parts.

### Step-by-Step Solution:

1. **Identify the parts for integration by parts**:
We will let:
- \( u = x \) (which we will differentiate)
- \( dv = e^{2x} \, dx \) (which we will integrate)

Therefore, we need to compute \( du \) and \( v \):
- \( du = dx \)
- \( v = \int e^{2x} \, dx = \frac{1}{2} e^{2x} \)

2. **Apply the integration by parts formula**:
The formula for integration by parts is:
\[ \int u \, dv = uv - \int v \, du \]
Substituting our values:
\[ \int x e^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx \]

3. **Evaluate the remaining integral**:
We need to compute \( \int \frac{1}{2} e^{2x} \, dx \):
\[ \int \frac{1}{2} e^{2x} \, dx = \frac{1}{2} \cdot \frac{1}{2} e^{2x} = \frac{1}{4} e^{2x} \]

4. **Combine the results**:
Now substituting back into our equation:
\[ \int x e^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + C \]

5. **Factor out \( e^{2x} \)**:
We can factor \( e^{2x} \) out of the expression:
\[ \int x e^{2x} \, dx = e^{2x} \left( \frac{1}{2} x - \frac{1}{4} \right) + C \]

6. **Identify \( f(x) \)**:
From the expression \( e^{2x} f(x) + C \), we can identify:
\[ f(x) = \frac{1}{2} x - \frac{1}{4} \]

### Final Answer:
Thus, the function \( f(x) \) is:
\[ f(x) = \frac{1}{2} x - \frac{1}{4} \]