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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"):}`
Solution
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"):}`
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