The problem states that if $f$ is a continuous function that satisfies $\int_0^x f(t) dt = x^2(1+x)$, we need to find the value of $f(2)$.

1. Problem Description

The problem states that if $f$ is a continuous function that satisfies $\int_0^x f(t) dt = x^2(1+x)$, we need to find the value of $f(2)$.

2. Solution Steps

We are given that

$$\int_0^x f(t) dt = x^2(1+x)$$

Using the Fundamental Theorem of Calculus, we can differentiate both sides with respect to $x$.

$$\frac{d}{dx} \int_0^x f(t) dt = \frac{d}{dx} [x^2(1+x)]$$

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

$$f(x) = 2x + 3x^2$$

Now we want to find $f(2)$.

$$f(2) = 2(2) + 3(2^2) = 4 + 3(4) = 4+12 = 16$$

3. Final Answer

The final answer is 16.