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