Question

Prove that: 

`{:(int_(-a)^a f(x) dx  = 2 int_0^a f(x) dx",", "If"  f(x)  "is an even function"),(                                       = 0",", "if"  f(x)  "is an odd function"):}`

Answer 1

L.H.S becomes

`int_(-a)^a f(x)  dx = int_(-a)^0 f(x) dx + int_0^a  f(x) dx`   .......(i)

Consider `int_(-a)^0 f(x) dx`

Put x = – t

∴ dx = – dt

When x = – a, t = a

and when x = 0, t = 0

∴ `int_(-a)^0 f(x) dx = int_a^0 f(-t)(-dt)`

= `-int_a^0 f(-t) dt`

= `int_0^a f(-t) dt`    ......`[∵ int_a^b f(x) dx = - int_b^a f(x) dx]`

= `int_0^a f(-x) dx`    ......`[∵ int_a^b f(x) dx = int_a^b f(t) dt]`

Equation (i) becomes

`int_(-a)^a f(x) dx = int_0^a f(-x) dx + int_0^a f(x) dx`

= `int_0^a [f(-x) + f(x)] dx` ......(ii)

Case I:

If f(x) is an even function, then f(– x) = f(x),

Equation (ii) becomes

`int_(-a)^a f(x) dx = 2* int_0^a f(x) dx`

Case II:

If f(x) is an odd function, then f(– x) = – f(x),

Equation (ii) becomes

`int_(-a)^a f(x) dx` = 0

`{:(int_(-a)^a f(x) dx = 2* int_0^a f(x) dx",", "If"  f(x)  "is an even function"),(                                    = 0",", "if"  f(x)  "is an odd function"):}`