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

Let `Iint_(0)^(a)f(x)dx+int_(0)^(a)f(2a-x)dx`
`=I_(1)+I_(2)`
for `I_(2)`, put `2a-x=t`
`:.-dx=dt`
`:.dx=-dt`
when `x=0impliest=2a` and when `x=aimpliest=a`
`I_(2)=int_(0)^(a)f(2a-x).dx`
`=-int_(2a)^(a)f(t).dt`
`:.I=I_(1)+I_(2)`
`=int_(0)^(a)f(x).dx-int_(2a)^(a)f(t).dt`
`=int_(0)^(a)f(x)dx+int_(a)^(2a)f(t)dt`
`("":'int_(0)^(b)f(x)dx=-int_(b)^(a)f(x)dx)`
`:'int_(0)^(a)f(x).dx+int_(a)^(2a)f(x)dx`
`[:'int_(0)^(a)f(x)dx=int_(0)^(a)f(t)dt]`
`=int_(0)^(2a)f(x).dx`
`[:'int_(a)^(b)f(x).dx=int_(a)^(c)f(x).dx]`
`:.int_(0)^(2a)f(x).dx=int_(a)^(a)f(x)dx+int_(0)^(a)f(2a-x).dx`
Hence Proved.