Question

Evaluate : `int_3^9 [root(3)(12-x)]/[ root(3)(x) + root(3)(12 - x)]`

Answer 1

Let, I = `int_3^9 [root(3)(12-x)]/[ root(3)(x) + root(3)(12 - x)]`        ...(1)

By using the property

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

∴ I = `int_3^9 [root(3)( 12 - ( 3 + 9 - x ))]/[root(3)( 3 + 9 - x) + root(3)( 12 - ( 3 + 9 - x)) ]`

`∴ I = int_3^9 root(3)(x)/[root (3)(12 - x) + root(3)(x)]`dx    ...(2)

Adding (1) and (2)

∴ 2I = `int_3^9 [root(3)(12 - x) + root(3)(x)]/[root(3)(x) + root(3)(12-x)]` dx

∴ 2I = `int_3^9 1 dx`

∴ 2I = `[x]_3^9`

∴ 2I = 9 - 3

∴ 2I = 6

∴ I = `6/2` 

∴ I = 3.