If `a lt b lt c in R` and `f (a + b +c - x) = f (x)` for all, x then `int_(a)^(b) (x(f(x) + f(x + c))dx)/(a + b)` is:

Options

A. `int_(a - c)^(b - c) f (x + c)`
B. `int_(a + c)^(b + c) f (x + c)`
C. `int_(a)^(b) f (x + c)`
D. `int_(a + c)^(b + c) f (x)`

Answer

A

Explanation

`I = int_(a)^(b) x (f (x) + f (x + c)) dx`
`I = int_(a)^(b) x f (x) + int_(a)^(b) (a + b - x) f (a + b + c - x)`
`I = int_(a)^(b) x f (x) + int_(a)^(b) (a + b - x) f (x)`
`I = int_(a)^(b) x f (x) - int_(a)^(b) f (x) x + int_(a)^(b) (a + b) f (x)`
`I = int_(a)^(b) (a + b) f (x) = (a + b) int_(a)^(b) f (x)`
`I = (a + b) int_(a - c)^(b -c) f (x + c)`