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If | inta ^b f(x) dx| = inta ^b |f(x)| d...

If `| int_a ^b f(x) dx| = int_a ^b |f(x)| dx, a lt b`, then `f(x) = 0` has

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`" Let " I= int_(1)^(b) x f(x) dx" "...(1)`
`rArr I= int_(a)^(b) (a+b-x)f(a+b-x)dx`
`=int_(a)^(b) (a+b-x)f(x) dx" "....(2)`
`[ :' f(a+b-x)=f(x)" given "]`
Adding equations (1) and (2)
`rArr 2I = int_(a)^(b) (a+b) f(x) dx`
` I=(a+b)/(2)int_(a)^(b) f(x) dx`
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