Let f : - π 2 , π 2 → R be a differentiable function such that f 0 = 1 2 , If lim x → 0 x ∫ 0 x f ( t ) d t e x 2 - 1 = α , then 8 α 2 is equal to :

Step-by-step solution

Let, y = lim x → 0 x ∫ 0 x f ( t ) d t e x 2 - 1 ⇒ y = lim x → 0 x ∫ 0 x f ( t ) d t e x 2 - 1 x 2 × x 2 We know that, lim x → 0 e x 2 - 1 x 2 = 1 ⇒ y = lim x → 0 ∫ 0 x f ( t ) d t x Applying L-hospital's rule and Newton Leibnitz Theorem we get, ⇒ y = lim x → 0 f ( x ) 1 ⇒ y = f 0 It is given that, f ( 0 ) = 1 2 ⇒ y = 1 2 ⇒ α = 1 2 ⇒ 8 α 2 = 2