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Question
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
Options
`(a + b)/2 int_a^b f(b - x) dx`
`(a + b)/2 int_a^b f(b + x) dx`
`(b - a)/2 int_a^b f(x) dx`
`(a + b)/2 int_a^b f(x) dx`
Solution
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to `underline((a + b)/2 int_a^b f(x) dx)`.
Explanation:
Let `I = int_a^b x f(x) dx`
`= int_a^b (a + b - x) f(a + b - x) dx` ` ...[because int_a^b f(x) dx = int_a^b f(a + b - x)] dx`
`I = int_a^b f(a + b - x) f(x) dx` ` ... [because f(a + b - x) = f(x) "Given"]`
∴ `I = int_a^b [(a + b) f(x) - x f(x)]dx`
`= (a + b) int_a^b f(x) dx - int_a^b x f(x) dx`
= `(a + b) int_a^b f(x) dx - 1`
∴ `2I = (a + b) int_a^b f(x) dx`
∴ `I = (a + b)/2 int_a^b f(x) dx`
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