Final answer:
To find an explicit formula for f(x), differentiate both sides of the equation using the Fundamental Theorem of Calculus. The derivative of the integral is the original function, so f(x) is found by differentiating the left side. The derivative of the right side gives x^2/2 + x^3/2 + 2x^3/8 as the explicit formula for f(x). To determine the value of C, substitute x = 0 into the equation and solve for C, which results in C = 0.
Step-by-step explanation:
To find an explicit formula for f(x), we can differentiate both sides of the given equation with respect to x. This will give us the derivative of the integral on the left side and the derivative of the right side.
By applying the Fundamental Theorem of Calculus, we know that the derivative of an integral is the original function. Therefore, the derivative of Int 0 to x f(t)dt with respect to x is f(x).
On the other side, the derivative of int x to 1 t^2f(t)dt + x^2/4 + x^4/8 + C with respect to x will give us the derivative of each term, which results in x^2/2 + x^3/2 + 2x^3/8.
Therefore, we have found the explicit formula for f(x) as f(x) = x^2/2 + x^3/2 + 2x^3/8.
To determine the value of C, we can plug in a value of x into the given equation and solve for C. Let's choose x = 0 because it simplifies the equation.
When we substitute x = 0 into the equation, we get Int 0 to 0 f(t)dt = int 0 to 1 t^2f(t)dt + 0^2/4 + 0^4/8 + C. The integral on the left side is zero, and the right side simplifies to C.
Therefore, the value of C is 0.