Let F : 3 , 5 → R be a twice differentiable function on 3 , 5 such that F x = e - x ∫ 3 x 3 t 2 + 2 t + 4 F ' t d t . If F ' 4 = α e β - 224 e β - 4 2 , then α + β is equal to _____.

Step-by-step solution

Given, F x = e - x ∫ 3 x 3 t 2 + 2 t + 4 F ' t d t . So, F 3 = e - 3 ∫ 3 3 3 t 2 + 2 t + 4 F ' t d t . ⇒ F 3 = 0 Again, e x F x = ∫ 3 x 3 t 2 + 2 t + 4   F ' t d t Differentiate both sides ⇒ e x F x + e x F ' x = 3 x 2 + 2 x + 4   F ' x (Newton Leibnitz Theorem for Definite Integral) ⇒ e x - 4 d y d x + e x y = 3 x 2 + 2 x (Use y = F x ) ⇒ d y d x + e x e x - 4 y = 3 x 2 + 2 x e x - 4 Here, integrating factor is e ∫ e x e x - 4 d x = e ln e x