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If f (6-x ) =f (x), for all then 1/5 int...

If `f (6-x ) =f (x),` for all then `1/5 int _(2)^(3) x [f (x) + f (x+1)]dx` is equal to :

A

`int _(3) ^(4) f (x+z) dx`

B

`int _(3 )^(4) f (x+1) dx`

C

` int _(1) ^(2) f (x+1) dx`

D

`int _(1) ^(3) f (x) dx`

Text Solution

Verified by Experts

The correct Answer is:
C
`I =1/5 int _(2)^(3) x [ f (x) + f (x+1)]dx" "…(i)`
`I =1/5 int _(2)^(3) ( 5-x) [f (5-x) + f (6-x)]dx` Adding (i) and (ii)
`2I =5/5 int _(2)^(3) [f (x) + f(x+1) ]dx`
`2I = int _(2)^(2) f (x) dx + int _(2) ^(3) f (x+1) dx`
`2I = int _(2)^(3) f (x) dx + int _(2)^(3) f [6-x] dx`
`2I = int _(2) ^(3) f (x) dx + int _(2)^(3) f (x) dx`
`I = int _(2)^(3) f (x) dx implies I= int _(1) ^(2) f (x +1) dx`
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