Let f : 0 , ∞ → R and F x = ∫ 0 x t f t d t . If F x 2 = x 4 + x 5 , then ∑ r = 1 12 f r 2 is equal to:

Step-by-step solution

Given: F x = ∫ 0 x t ⋅ f t d t Now, applying Newton Leibnitz Theorem we get, F ' x = x ⋅ f x And F x 2 = x 4 + x 5 Let x 2 = t ⇒ F t = t 2 + t 5 2 ⇒ F ' t = 2 t + 5 2 t 3 2 as F ' x = x ⋅ f x ⇒ t ⋅ f t = 2 t + 5 2 t 3 2 ⇒ f t = 2 + 5 2 t 1 2 ⇒ ∑ r = 1 12 f r 2 = ∑ r = 1 12 2 + 5 2 r ⇒ ∑ r = 1 12 f r 2 = 24 + 5 2 12 13 2 ⇒ ∑ r = 1 12 f r 2 = 219