Let f(x) = _ n ( x 2 )^ 2n (x + 10) + a x^2 + c x + b ( x 2 )^ 2n + 1 , where a, b, c R and n N . If _ x 2 f(x) and _ x -2 f(x) both exist, and _ -2 ^2 f(x) dx = 8 , then the value of 2 ₁^4 f(x) dx is equal to

Step-by-step solution

The function f(x) can be simplified based on the value of | x 2 | . For |x| For |x| > 2 , ( x 2 )^ 2n as n , so f(x) = x + 10 . Since _ x 2 f(x) exists, the LHL and RHL at x = 2 must be equal: _ x 2^- (a x^2 + c x + b) = _ x 2^+ (x + 10) 4a + 2c + b = 12 --- (1) Since _ x -2 f(x) exists, the LHL and RHL at x = -2 must be equal: _ x -2^- (x + 10) = _ x -2^+ (a x^2 + c x + b) 8 = 4a - 2c + b --- (2) Subtracting (2) from (1) gives 4c = 4 c = 1 . Adding (1) and (2) gives 8a + 2b = 20 4a + b = 10 --- (3) We are given _