The area bounded by the curve `y=f(x)` the coordinate axes and the line x = t is given by `te^(t)` then f(x) =

Options

A. `e^(x)(x+1)`
B. `e^(x)(x-1)`
C. `x(1+e^(x))`
D. `x(1-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 \( x = t \) is equal to \( t e^t \).

### Step-by-Step Solution:

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

2. **Setting Up the Equation**:
We can set up the equation:
\[ \int_0^t f(x) \, dx = t e^t \]

3. **Differentiating Both Sides**:
To find \( f(t) \), we differentiate both sides with respect to \( t \):
\[ \frac{d}{dt} \left( \int_0^t f(x) \, dx \right) = \frac{d}{dt} (t e^t) \]
By the Fundamental Theorem of Calculus, the left side simplifies to:
\[ f(t) - f(0) \]
The right side can be differentiated using the product rule:
\[ \frac{d}{dt} (t e^t) = e^t + t e^t \]
Therefore, we have:
\[ f(t) - f(0) = e^t + t e^t \]

4. **Assuming \( f(0) = 0 \)**:
Since \( f(0) \) represents the value of the function at \( x = 0 \), we can assume \( f(0) = 0 \) (as it is a common assumption in such problems). Thus, we have:
\[ f(t) = e^t + t e^t \]

5. **Factoring the Expression**:
We can factor out \( e^t \):
\[ f(t) = e^t (1 + t) \]

6. **Final Expression**:
Since \( t \) is a dummy variable, we can replace \( t \) with \( x \) to express \( f(x) \):
\[ f(x) = e^x (1 + x) \]

### Conclusion:
The function \( f(x) \) is given by:
\[ f(x) = e^x (1 + x) \]