`If f(x) =x + int_(0)^(1) (xy^(2)+x^(2)y)(f(y)) dy, "find" f(x)` if x and y are independent.

Answer

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

\[ f(x) = x + \int_{0}^{1} (xy^2 + x^2y) f(y) \, dy \]

### Step 1: Simplify the Integral
We can separate the integral into two parts:

\[ f(x) = x + x \int_{0}^{1} y^2 f(y) \, dy + x^2 \int_{0}^{1} y f(y) \, dy \]

Let’s denote:

\[ k = \int_{0}^{1} y^2 f(y) \, dy \quad \text{and} \quad c = \int_{0}^{1} y f(y) \, dy \]

Then we can rewrite \(f(x)\) as:

\[ f(x) = x + xk + x^2c \]

### Step 2: Factor Out x
Now we can factor out \(x\):

\[ f(x) = x(1 + k) + cx^2 \]

### Step 3: Identify the Form of f(x)
This suggests that \(f(x)\) is a quadratic function of the form:

\[ f(x) = Ax + Bx^2 \]

where \(A = 1 + k\) and \(B = c\).

### Step 4: Substitute f(y) Back into the Integrals
Now, substituting \(f(y) = Ay + By^2\) into the definitions of \(k\) and \(c\):

1. For \(k\):

\[ k = \int_{0}^{1} y^2 (Ay + By^2) \, dy = A \int_{0}^{1} y^3 \, dy + B \int_{0}^{1} y^4 \, dy \]

Calculating the integrals:

\[ \int_{0}^{1} y^3 \, dy = \frac{1}{4}, \quad \int_{0}^{1} y^4 \, dy = \frac{1}{5} \]

Thus,

\[ k = A \cdot \frac{1}{4} + B \cdot \frac{1}{5} \]

2. For \(c\):

\[ c = \int_{0}^{1} y (Ay + By^2) \, dy = A \int_{0}^{1} y^2 \, dy + B \int_{0}^{1} y^3 \, dy \]

Calculating the integrals:

\[ \int_{0}^{1} y^2 \, dy = \frac{1}{3} \]

Thus,

\[ c = A \cdot \frac{1}{3} + B \cdot \frac{1}{4} \]

### Step 5: Set Up the System of Equations
Now we have two equations:

1. \(k = \frac{A}{4} + \frac{B}{5}\)
2. \(c = \frac{A}{3} + \frac{B}{4}\)

Substituting \(k\) and \(c\) back into the expression for \(f(x)\):

\[ f(x) = (1 + k)x + cx^2 \]

### Step 6: Solve the System
We can substitute \(k\) and \(c\) into the equations and solve for \(A\) and \(B\).

After solving the equations, we find:

\[ A = \frac{180}{119}, \quad B = \frac{80}{119} \]

### Final Step: Write the Final Form of f(x)
Thus, we can express \(f(x)\) as:

\[ f(x) = \frac{180}{119} x + \frac{80}{119} x^2 \]

### Conclusion
The final answer is:

\[ \boxed{f(x) = \frac{180}{119} x + \frac{80}{119} x^2} \]