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`

Answer 1

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`