The area bounded by the curve `y=f(x)` (where `f(x) geq 0`), the co-ordinate axes & the line `x=x_1` is given by `x_1.e^(x_1)`. Therefore `f(x)` equals

Options

A. `e^(x)`
B. `xe^(x)`
C. `xe^(x)-e^(x)`
D. `xe^(x)+e^(x)`

Answer

To solve the problem, we need to find the function \( f(x) \) given that the area bounded by the curve \( y = f(x) \), the coordinate axes, and the line \( x = x_1 \) is equal to \( x_1 e^{x_1} \).

### Step-by-Step Solution:

1. **Understand the Area Expression**:
The area \( A \) bounded by the curve \( y = f(x) \), the x-axis, and the line \( x = x_1 \) can be expressed as:
\[ A = \int_0^{x_1} f(x) \, dx \]
According to the problem, this area is also given by:
\[ A = x_1 e^{x_1} \]

2. **Set Up the Equation**:
We can set the two expressions for the area equal to each other:
\[ \int_0^{x_1} f(x) \, dx = x_1 e^{x_1} \]

3. **Differentiate Both Sides**:
To find \( f(x) \), we differentiate both sides with respect to \( x_1 \). We will use the Fundamental Theorem of Calculus on the left side:
\[ \frac{d}{dx_1} \left( \int_0^{x_1} f(x) \, dx \right) = f(x_1) \]
For the right side, we apply the product rule:
\[ \frac{d}{dx_1} (x_1 e^{x_1}) = e^{x_1} + x_1 e^{x_1} \]

4. **Equate the Derivatives**:
Now we equate the derivatives from both sides:
\[ f(x_1) = e^{x_1} + x_1 e^{x_1} \]

5. **Change Variable**:
Since we have \( f(x_1) \) and we want \( f(x) \), we can simply replace \( x_1 \) with \( x \):
\[ f(x) = e^{x} + x e^{x} \]

6. **Final Answer**:
Thus, the function \( f(x) \) is:
\[ f(x) = (x + 1)e^{x} \]