The value of ∫_2^10(√(12 - x))/(√x + √(12 - x)) dx is

The correct answer is Option (4) → 4

Let

    \( \displaystyle I = \int_{2}^{10}\frac{\sqrt{12 - x}}{\sqrt{x} + \sqrt{12 - x}}\,dx \)

Here a + b = 2 + 10 = 12. By the property \(\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx\), replace x by 12 − x:

    \( \displaystyle I = \int_{2}^{10}\frac{\sqrt{x}}{\sqrt{12 - x} + \sqrt{x}}\,dx \)

Adding the two forms of I, the numerators add up to the denominator:

    \( \displaystyle 2I = \int_{2}^{10}\frac{\sqrt{x} + \sqrt{12 - x}}{\sqrt{x} + \sqrt{12 - x}}\,dx \)

    \( \displaystyle = \int_{2}^{10} 1\,dx \)

    \( = 10 - 2 = 8 \)

So

    \(I = 4\)