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. `overset(b-c)underset(a-c)int f(x +c)`
B. `overset(b+c)underset(a+c)int f(x +c)`
C. `underset(a)overset(b)int f(x+c)`
D. `underset(a+c)overset(b+c) intf(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) underset(a-c)overset(b-c) int f(x+c)`