Question

Let F(x) = ∫2cos²tdt for t∈[x, x² + π/6] for all x ∈ R and f: [0, 1/2]→ [0, ∞] be a continuous function. For a ∈ [0, 1/2], if F'(a) + 2 is the area of the region bounded by x = 0, y = 0, y = f(x) and x = a,  then f(0) is _____. 

Answer 1

We have

Differentiating both sides, we get