Prove that: `int_0^(2a)f(x)dx=int_0^(2a)f(2a-x)dxdot`

Explanation

`L.H.S. = ∫_0^(2a)​f(x)dx`

By property, the integral limits can be split from `0` to `a` and `a` to `2a`

∴ L.H.S.
`=∫_0^(a)f(x)dx+∫_a^(2a)​f(x)dx`

In the second integral, substitute
`x=2a−t`, then `dx=−dt`

`L.H.S. =∫_0^(a)​f(x)dx+∫_a^0​`
`f(2a−t)(−dt)`
`=∫_0^a​f(x)dx−∫_a^0​f(2a−t)dt`

Interchanging the limits of the second integral
and changing the variable t to x, we get

`L.H.S. =∫_0^a​f(x)dx+∫_0^a​f(2a−x)dx`

...