If `int x e ^(2x) `is equal to `e ^(2x) f (x) + c` where C is constant of integration then `f (x) is ((2x -1))/( 2)`

Answer

To solve the problem, we need to evaluate the integral \( \int x e^{2x} \, dx \) and express it in the form \( e^{2x} f(x) + C \), where \( C \) is the constant of integration. We will use integration by parts to achieve this.

### Step-by-Step Solution:

1. **Set Up Integration by Parts**:
We will use the integration by parts formula:
\[ \int u \, dv = u v - \int v \, du \]
Here, we can choose:
- \( u = x \) (which means \( du = dx \))
- \( dv = e^{2x} dx \) (which means \( v = \frac{1}{2} e^{2x} \))

**Hint**: Choose \( u \) to be the polynomial part and \( dv \) to be the exponential part for easier integration.

2. **Apply the Integration by Parts Formula**:
Now substitute \( u \) and \( v \) into the integration by parts formula:
\[ \int x e^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx \]

3. **Integrate the Remaining Integral**:
The remaining integral \( \int \frac{1}{2} e^{2x} \, dx \) can be computed as follows:
\[ \int \frac{1}{2} e^{2x} \, dx = \frac{1}{2} \cdot \frac{1}{2} e^{2x} = \frac{1}{4} e^{2x} \]

4. **Combine the Results**:
Substitute back into the 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. **Express in Terms of \( f(x) \)**:
From the expression, we can identify \( f(x) \):
\[ f(x) = \frac{1}{2} x - \frac{1}{4} \]
To express this in a more standard form, we can multiply through by 2:
\[ f(x) = \frac{2x - 1}{4} \]

### Conclusion:
The statement that \( f(x) = \frac{2x - 1}{2} \) is **false**. The correct form is \( f(x) = \frac{2x - 1}{4} \).