If `f(x)` and `g(x)` are continuous functions satisfying `f(x) = f(a-x)` and `g(x) + g(a-x) = 2`, then what is `int_(0)^(a) f(x) g(x)dx` equal to ?

Options

A. `int_(0)^(a) g(x) dx`
B. `int_(0)^(a) f(x) dx`
C. `2int_(0)^(a) f(x) dx`
D. `0`

Answer

B

Explanation

`I = int_(0)^(a) f(x) . g(x) dx`
`I= int_(0)^(a) f(a-x) .g(a-x) dx`
`I = int_(0)^(a) f(x).[2-g(x)] dx`
`:. f(a-x) = f(x)` and `g(a-x) = 2g(x)`
`I = int_(0)^(a) 2.f(x).dx - int_(0)^(a) f(x) .g(x).dx`
`I = 2.int_(0)^(a)f(x) - 1`
`rArr cancel2I = cancel2int_(@)^(a) f(x) dx`
`:. I = int_(0)^(a) f(x) dx`