1) Let f:[0,2]→ R be a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1.Let F(x)=∫x20f(√t)dt,forx∈[0,2] . If F'(x)= f '(x) for all x ε (0,2) , then F(2) equal to A) e2−1 B) e4−1 C) e-1 D) e4 Answer: Option BExplanation: Plan Newton - Leibnitz.s formula ddx[∫ψ(X)ϕ(X)f(t)dt]=f{ψ(X)}{ddxψ(X)}−f(ϕ(X)){ddxϕ(X)} Given, F(X)=∫x20f(√t)dt ∴ F'(x)= 2x f(x) Also, F '(x)= f' (x) ⇒ 2xf(x)=f′(x)⇒f′(x)f(x)=2x ⇒ ∫f′(x)f(x)dx=∫2xdx ⇒ ln f(x)= x2+ c ⇒ f(x)=ex2+c ⇒ f(x)=Kex2 [K=ec] Now, f(0)=1 1=K Hence f(x)=ex2 F(2)=∫40etdt =[et]40=e4−1