Step-by-Step Solution
Key Concept: Applying Leibniz rule for differentiation under the integral sign to find the value of the function at a specific point.
Differentiating using Leibniz rule, we get $\frac{d}{dy}[F(x+a)^2 - x^2f(x^2)]dx = 2\cos(2x)x$. Putting $x = 1$: $4f(1) - 2f(1) = 2\pi$, which gives $2f(1) = 2\pi$, so $f(1) = \pi$.
Correct Answer: 4