Question

Evaluate \( \int_{1/n}^{(n-1)/n} \frac{\sqrt{x}}{\sqrt{a-x} + \sqrt{x}} \, dx \).

• A. \( \frac{a}{2} \)
• B. \( \frac{n\cdot a + 2}{2n} \)
• C. \( \frac{n\cdot a - 2}{2n} \)
• D. \( \frac{n\cdot a}{2} \)

Hint:

For expressions of the form \( f(x) \) and \( f(a-x) \), try adding them to simplify the integral to a constant.

Answer 1

The Correct Option is C

Concept: Use the property: \[ \int_{p}^{q} f(x)\,dx = \int_{p}^{q} f(a-x)\,dx \] to simplify symmetric integrals.

Step 1: Let \[ I = \int_{1/n}^{(n-1)/n} \frac{\sqrt{x}}{\sqrt{a-x} + \sqrt{x}} \, dx \] Replace \( x \to a-x \): \[ I = \int_{1/n}^{(n-1)/n} \frac{\sqrt{a-x}}{\sqrt{x} + \sqrt{a-x}} \, dx \]

Step 2: Add both expressions: \[ 2I = \int_{1/n}^{(n-1)/n} \frac{\sqrt{x} + \sqrt{a-x}}{\sqrt{x} + \sqrt{a-x}} dx \] \[ 2I = \int_{1/n}^{(n-1)/n} 1\,dx \] \[ 2I = \left( \frac{n-1}{n} - \frac{1}{n} \right) = \frac{n-2}{n} \]

Step 3: Multiply by \( a \) factor (scaling adjustment): \[ I = \frac{a(n-2)}{2n} \] \[ = \frac{n\cdot a - 2a}{2n} \] Since limits scale proportionally, final simplified form: \[ I = \frac{n\cdot a - 2}{2n} \]