Let `f:(0, oo) rarr R and F(x^(2))=int_(0)^(x^(2))f(t)dt. " If "F(x^(2))=x^(2)(1+x)`, then f(4) is equal to ________________

Answer

To solve the problem, we need to find the function \( f(t) \) given that:

\[ F(x^2) = \int_0^{x^2} f(t) \, dt = x^2(1 + x) \]

We will differentiate both sides with respect to \( x \) to find \( f(t) \).

### Step 1: Differentiate \( F(x^2) \)

Using the Fundamental Theorem of Calculus and the chain rule, we differentiate the left-hand side:

\[ \frac{d}{dx} F(x^2) = \frac{d}{dx} \left( \int_0^{x^2} f(t) \, dt \right) = f(x^2) \cdot \frac{d}{dx}(x^2) = f(x^2) \cdot 2x \]

### Step 2: Differentiate the right-hand side

Now, we differentiate the right-hand side:

\[ \frac{d}{dx} (x^2(1 + x)) = \frac{d}{dx} (x^2 + x^3) = 2x + 3x^2 \]

### Step 3: Set the derivatives equal to each other

Now we set the derivatives from both sides equal to each other:

\[ f(x^2) \cdot 2x = 2x + 3x^2 \]

### Step 4: Solve for \( f(x^2) \)

To isolate \( f(x^2) \), we divide both sides by \( 2x \) (assuming \( x \neq 0 \)):

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

### Step 5: Substitute \( x^2 = 4 \) to find \( f(4) \)

Since we need to find \( f(4) \), we set \( x^2 = 4 \), which gives \( x = 2 \):

\[ f(4) = 1 + \frac{3}{2}(2) = 1 + 3 = 4 \]

Thus, the value of \( f(4) \) is:

\[ \boxed{4} \]