If `f(x)` is a continuous function defined on `[0,\ 2a]dot\ ` Then prove that `int_0^(2a)f(x)dx=int_0^a{f(x)+(2a-x)}dx`

Explanation

We have to proof `int_0^(2a)f(x)dx=int_0^a{f(x)+(2a-x)}dx`

Take , L.H.S

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

As, `int_a ^b f(x) dx=int_a ^c f(x) dx+ int_c ^b f(x) dx `

`I=int_0^a f(x) dx+ int_a ^2a f(x) dx `

`I=I_1+I_2`

`I_2=int_a^2a f(x) dx`

Let `2a-x=t implies x=2a-t implies dx=-dt`
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