Prove the following properties of definite integrals :
`int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx`

Explanation

We shall use the following results :
`int_(a)^(b)f(x)dx=int_(a)^(b)f(t)dt" …(1)"`
and `int_(a)^(b)f(x)dx=-int_(b)^(a)f(x)dx" …(2)"`
Consider `int_(0)^(a)f(x)dx`
Put `x=a-t,` Then `dx=-dt`
When `x=0, a-t =0 " "therefore" "t=a.` When `x = a, a-t =a" "therefore t =0`
`therefore" "int_(0)^(a)f(x)dx=int_(a)^(0)f(a-t)(-dt)=-int_(a)^(0)f(a-t)dt`
`=int_(0)^(a)f(a-t)dt`
`=int_(0)^(a)f(a-x)dx`
`therefore int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx`