If `f` is an integrable function such that `f(2a-x)=f(x),` then prove that `int_0^(2a)f(x)dx=2int_0^af(x)dx`

Explanation

Given: `f(2a-x)=f(x)....(i)`

We have to proof `int_0^(2a)f(x)dx=2int_0^af(x)dx`

`int_0^(2a)f(x)dx=int_0^a[f(x)+f(2a-x)]dx`

From(i)

`int_0^(2a)f(x)dx=int_0^a[f(x)+f(x)]dx`

`int_0^(2a)f(x)dx=int_0^a[f(x)+f(x)]dx`

`int_0^(2a)f(x)dx=2int_0^af(x)dx`

Hence Proved