Question

Let f:(0, ∞) → R and f(x) = ∫f(t)dt for t∈ [0, x]. if f(x²) = x²(1 + x), then find f(4)?

Answer 1

By definition of f(x) we have

Differentiate both sides,

f(x²).2x + 0 = 2x + 3x²

Put

x = 2 ⇒ 4f(4) = 16 ⇒ f(4) = 4