If `int_(0)^(x)f(t)dt=e^(x)-ae^(2x)int_(0)^(1)f(t)e^(-t)dt`, then

Options

A. `a=(1)/(3-2e)`
B. `f(x)=e^(x)-2e^(2x)`
C. `a=(1)/(e)`
D. `f(x)=e^(x)-e^(-x)`

Explanation

To solve the problem, we start with the given equation:

\[ \int_{0}^{x} f(t) dt = e^{x} - ae^{2x} \int_{0}^{1} f(t)e^{-t} dt \]

### Step 1: Evaluate the integral at \( x = 0 \)

Substituting \( x = 0 \):

\[ \int_{0}^{0} f(t) dt = e^{0} - ae^{0} \int_{0}^{1} f(t)e^{-t} dt \]

The left-hand side becomes:

\[ 0 = 1 - a \int_{0}^{1} f(t)e^{-t} dt \]

This simplifies to:

\[ a \int_{0}^{1} f(t)e^{-t} dt = 1 \]

Thus, we can express the integral as:

\[ \int_{0}^{1} f(t)e^{-t} dt = \frac{1}{a} \]

### Step 2: Substitute back into the original equation

Now, substituting this back into the original equation:

\[ \int_{0}^{x} f(t) dt = e^{x} - ae^{2x} \cdot \frac{1}{a} \]

This simplifies to:

\[ \int_{0}^{x} f(t) dt = e^{x} - e^{2x} \]

### Step 3: Differentiate both sides with respect to \( x \)

Differentiating both sides gives:

\[ f(x) = \frac{d}{dx} \left( e^{x} - e^{2x} \right) \]

Calculating the derivative:

\[ f(x) = e^{x} - 2e^{2x} \]

### Conclusion

Thus, the function \( f(x) \) is:

\[ f(x) = e^{x} - 2e^{2x} \]