Property 6: If f(x) is a continuous function defined on `[0; 2a]` then ` int_0 ^2a = int_0 ^a f(x) dx + int_0 ^a f(2a - x) dx`

Explanation

We shall use the following results :
`int_(a)^(b)f(x)dx=int_(a)^(b)f(t)dt" …(1)"`
`int_(a)^(b) intf(x) dx=-int_(b)^(a)f(x)dx" …(2)"`
If c is between a and b, then
`int_(a)^(b)f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx" ...(3)"`
Since a lies between 0 and 2a, by (3), we have,
`I=int_(0)^(2a)f(x)dx=int_(0)^(a)f(x)dx+int_(a)^(2a)f(x)dI_(1)+I_(2)" ...(Say)"`
In `I_(2),` put `x=2a-t`, then `dx=-dt`
When `x=a, 2a-t=a" "therefore" "t=a and" when "x=2a, 2a-t = 2a " "therefore" "t=0`
`therefore int_(a)^(2a)f(x) dx=int_(a)^(0)f(2a-t)(-dt)=-int_(a)^(0)f(2a-t)dt`
`=int_(0)^(a)f(2a-t)dt" ...[By (2)]"`
`=int_(0)^(a)f(2a-x)dx" ...[By (1)]"`
`therefore" "int_(0)^(2a)f(x)dx=int_(0)^(a)f(x)dx+int_(0)^(a)f(2a-x)dx.`