Question

Find \( \int \sqrt{\frac{x + 2}{x - 2}} dx \)

Hint:

Rationalizing the numerator is a powerful technique for integrals of the form \( \sqrt{\frac{L_1}{L_2}} \).
Always check for the direct derivative of the term inside the square root in the numerator.

Answer 1

Concept:
• Rationalization of the integrand to remove the square root from the numerator.
• Integration of square root functions using standard formulas.

Step 1: Rationalize the numerator
Multiply the numerator and denominator by \( \sqrt{x + 2} \): \[ \int \sqrt{\frac{x + 2}{x - 2} \cdot \frac{x + 2}{x + 2}} dx = \int \frac{x + 2}{\sqrt{x^2 - 4}} dx \]

Step 2: Split the integral
\[ I = \int \frac{x}{\sqrt{x^2 - 4}} dx + \int \frac{2}{\sqrt{x^2 - 4}} dx \] Let these be \( I_1 \) and \( I_2 \).

Step 3: Evaluate each part
For \( I_1 = \int \frac{x}{\sqrt{x^2 - 4}} dx \), put \( x^2 - 4 = t \), so \( 2x \, dx = dt \): \[ I_1 = \frac{1}{2} \int \frac{dt}{\sqrt{t}} = \frac{1}{2} [2\sqrt{t}] = \sqrt{x^2 - 4} \] For \( I_2 = \int \frac{2}{\sqrt{x^2 - 4}} dx \), use the formula \( \int \frac{dx}{\sqrt{x^2 - a^2}} = \log |x + \sqrt{x^2 - a^2}| \): \[ I_2 = 2 \log |x + \sqrt{x^2 - 4}| \]

Step 4: Combine the terms
\[ I = \sqrt{x^2 - 4} + 2 \log |x + \sqrt{x^2 - 4}| + C \]