Question

If f(a + b − x) = f(x), then `int_a^b x f(x) dx` is equal to ______.

विकल्प

• `(a + b)/2 int_a^b f(b - x) dx`

• `(a + b)/2 int_a^b f(a - 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 `underlinebb((a + b)/2 int_a^b f(x) dx)`.

Explanation:

Let I = `int_a^b x f(x) dx`  ...(1)

Using the property P₃:

`int_a^b f(x) dx = int_a^b f(a + b - x)dx`

∴ I = `int_a^b (a + b - x) f (a + b - x) dx`  ...[Given f(a + b − x) = f(x)]

I = `int_a^b (a + b - x) f(x) dx`

I = `int_a^b (a + b) f(x)dx - int_a^b x f(x) dx`  ....(2)

Adding (1) and (2), i.e., (1) + (2):

I + I = `int_a^b x f(x) dx + int_a^b (a + b) f(x)dx - int_a^b x f(x) dx`

2I = `int_a^b (a + b) f(x) dx`

2I = `(a + b) int_a^b f(x)dx`

∴ I = `(a + b)/2 int_a^b f(x)dx`