If `int_(0)^(x^(2)(1+x))f(t)dt=x`, then the value of `25f(2)` must be_________.

Answer

To solve the problem, we start with the equation:

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

We need to find the value of \( 25f(2) \).

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

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

\[ \frac{d}{dx} \left( \int_{0}^{x^2(1+x)} f(t) \, dt \right) = f(x^2(1+x)) \cdot \frac{d}{dx}(x^2(1+x)) \]

And the right-hand side:

\[ \frac{d}{dx}(x) = 1 \]

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

Now, we need to differentiate \( x^2(1+x) \):

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

### Step 3: Set up the equation

Now we substitute back into our differentiated equation:

\[ f(x^2(1+x)) \cdot (2x + 3x^2) = 1 \]

### Step 4: Solve for \( f(x^2(1+x)) \)

From the equation above, we can express \( f(x^2(1+x)) \):

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

### Step 5: Find \( f(2) \)

Next, we need to find \( f(2) \). We need to find \( x \) such that \( x^2(1+x) = 2 \):

Let \( x^2(1+x) = 2 \).

This simplifies to:

\[ x^3 + 2x^2 - 2 = 0 \]

### Step 6: Solve the cubic equation

We can try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Now, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

Next, try \( x = 1 \):

\[ 1^3 + 2(1^2) - 2 = 1 + 2 - 2 = 1 \quad \text{(not a root)} \]

### Step 7: Substitute \( x = 1 \) into \( f(x^2(1+x)) \)

We can substitute \( x = 1 \) into our expression for \( f \):

\[ f(1^2(1+1)) = f(2) = \frac{1}{2(1) + 3(1^2)} = \frac{1}{2 + 3} = \frac{1}{5} \]

### Step 8: Calculate \( 25f(2) \)

Now, we can find \( 25f(2) \):

\[ 25f(2) = 25 \cdot \frac{1}{5} = 5 \]

Thus, the value of \( 25f(2) \) is:

\[ \boxed{5} \]