If `int_(0)^(x)f(t)dt = x^(2)-int_(0)^(x^(2))(f(t))/(t)dt` then find `f(1)`.

Answer

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

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

### Step 1: Differentiate both sides with respect to \(x\)

Using the Fundamental Theorem of Calculus and the Leibniz rule for differentiation under the integral sign, we differentiate both sides:

\[ \frac{d}{dx} \left( \int_{0}^{x} f(t) \, dt \right) = f(x) \]

For the right-hand side, we differentiate:

\[ \frac{d}{dx} \left( x^2 - \int_{0}^{x^2} \frac{f(t)}{t} \, dt \right) = 2x - \frac{d}{dx} \left( \int_{0}^{x^2} \frac{f(t)}{t} \, dt \right) \]

Using the chain rule on the integral:

\[ \frac{d}{dx} \left( \int_{0}^{x^2} \frac{f(t)}{t} \, dt \right) = \frac{f(x^2)}{x^2} \cdot 2x \]

Putting it all together, we get:

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

### Step 2: Simplify the equation

Rearranging the equation gives:

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

### Step 3: Substitute \(x = 1\)

Now we want to find \(f(1)\):

\[ f(1) = 2 \cdot 1 - \frac{2 f(1^2)}{1} = 2 - 2 f(1) \]

### Step 4: Solve for \(f(1)\)

Rearranging gives:

\[ f(1) + 2 f(1) = 2 \]

\[ 3 f(1) = 2 \]

\[ f(1) = \frac{2}{3} \]

Thus, the final answer is:

\[ \boxed{\frac{2}{3}} \]