Let `f` be a differentiable function on `R` and satisfying the integral equation
`x int_(0)^(x)f(t)dt-int_(0)^(x)tf(x-t)dt=e^(x)-1 AA x in R`. Then `f(1)` equals to ___

Answer

To solve the given problem, we will follow these steps:

### Step 1: Write down the integral equation
We start with the integral equation given in the problem:
\[ x \int_{0}^{x} f(t) dt - \int_{0}^{x} t f(x - t) dt = e^{x} - 1 \]

### Step 2: Change the variable in the second integral
In the second integral, we can change the variable \( t \) to \( x - t \). Thus, we have:
\[ \int_{0}^{x} t f(x - t) dt = \int_{0}^{x} (x - u) f(u) du \]
where \( u = x - t \) and \( dt = -du \).

### Step 3: Rewrite the second integral
Substituting \( u \) back into the integral, we can express it as:
\[ \int_{0}^{x} t f(x - t) dt = \int_{0}^{x} (x - u) f(u) du = x \int_{0}^{x} f(u) du - \int_{0}^{x} u f(u) du \]

### Step 4: Substitute back into the original equation
Now, substituting this back into the original equation gives us:
\[ x \int_{0}^{x} f(t) dt - \left( x \int_{0}^{x} f(t) dt - \int_{0}^{x} t f(t) dt \right) = e^{x} - 1 \]
This simplifies to:
\[ \int_{0}^{x} t f(t) dt = e^{x} - 1 \]

### Step 5: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx} \left( \int_{0}^{x} t f(t) dt \right) = \frac{d}{dx} (e^{x} - 1) \]
Using the Leibniz rule for differentiation of integrals, we have:
\[ x f(x) = e^{x} \]

### Step 6: Solve for \( f(x) \)
From the equation \( x f(x) = e^{x} \), we can solve for \( f(x) \):
\[ f(x) = \frac{e^{x}}{x} \]

### Step 7: Find \( f(1) \)
Now, we need to find \( f(1) \):
\[ f(1) = \frac{e^{1}}{1} = e \]

### Conclusion
Thus, the value of \( f(1) \) is:
\[ \boxed{e} \]